Understanding Matrix Inversion

Oct 13, 2025 by miodeya.com 40 views

Introduction

In linear algebra, solving for a matrix X in a linear equation is a fundamental concept that involves finding the unknown matrix X that satisfies the given equation. In this article, we will focus on solving for the matrix X in the equation $\left[\begin{array}{cc}2 & -4 \\ 5 & 1\end{array}\right] X = \left[\begin{array}{cc}-14 & -14 \\ -13 & 9\end{array}\right]$ . We will use the method of matrix inversion to solve for the matrix X.

Understanding Matrix Inversion

Matrix inversion is a technique used to find the inverse of a matrix, which is denoted by $A^{-1}$ . The inverse of a matrix A is a matrix that, when multiplied by A, results in the identity matrix I. The identity matrix I is a square matrix with ones on the main diagonal and zeros elsewhere.

To find the inverse of a matrix A, we can use the following formula:

A^{-1} = \frac{1}{\det(A)} \times \text{adj}(A)

where $\det(A)$ is the determinant of the matrix A, and $\text{adj}(A)$ is the adjugate matrix of A.

Finding the Determinant of the Matrix A

The determinant of a 2x2 matrix A is given by:

\det(A) = ad - bc

where a, b, c, and d are the elements of the matrix A.

In this case, the matrix A is $\left[\begin{array}{cc}2 & -4 \\ 5 & 1\end{array}\right]$ . Therefore, the determinant of A is:

\det(A) = (2)(1) - (-4)(5) = 2 + 20 = 22

Finding the Adjugate Matrix of A

The adjugate matrix of A is a matrix that is obtained by taking the transpose of the matrix of cofactors of A.

The matrix of cofactors of A is given by:

\left[\begin{array}{cc}1 & -5 \\ 4 & 2\end{array}\right]

The transpose of this matrix is:

\left[\begin{array}{cc}1 & 4 \\ -5 & 2\end{array}\right]

Therefore, the adjugate matrix of A is:

\text{adj}(A) = \left[\begin{array}{cc}1 & 4 \\ -5 & 2\end{array}\right]

Finding the Inverse of the Matrix A

Now that we have the determinant and the adjugate matrix of A, we can find the inverse of A using the formula:

A^{-1} = \frac{1}{\det(A)} \times \text{adj}(A)

Substituting the values, we get:

A^{-1} = \frac{1}{22} \times \left[\begin{array}{cc}1 & 4 \\ -5 & 2\end{array}\right]

Simplifying, we get:

A^{-1} = \left[\begin{array}{cc}\frac{1}{22} & \frac{4}{22} \\ -\frac{5}{22} & \frac{2}{22}\end{array}\right]

A^{-1} = \left[\begin{array}{cc}\frac{1}{22} & \frac{2}{11} \\ -\frac{5}{22} & \frac{1}{11}\end{array}\right]

Solving for the Matrix X

Now that we have the inverse of the matrix A, we can solve for the matrix X using the equation:

AX = B

where A is the matrix $\left[\begin{array}{cc}2 & -4 \\ 5 & 1\end{array}\right]$ , X is the matrix we want to find, and B is the matrix $\left[\begin{array}{cc}-14 & -14 \\ -13 & 9\end{array}\right]$ .

Multiplying both sides of the equation by $A^{-1}$ , we get:

A^{-1}AX = A^{-1}B

Simplifying, we get:

X = A^{-1}B

Substituting the values, we get:

X = \left[\begin{array}{cc}\frac{1}{22} & \frac{2}{11} \\ -\frac{5}{22} & \frac{1}{11}\end{array}\right] \times \left[\begin{array}{cc}-14 & -14 \\ -13 & 9\end{array}\right]

Multiplying the matrices, we get:

X = \left[\begin{array}{cc}\frac{-14}{22} + \frac{-26}{22} & \frac{-14}{11} + \frac{18}{11} \\ \frac{70}{22} + \frac{-13}{22} & \frac{-70}{22} + \frac{9}{11}\end{array}\right]

Simplifying, we get:

X = \left[\begin{array}{cc}-\frac{40}{22} & -\frac{-4}{11} \\ \frac{57}{22} & \frac{-61}{22}\end{array}\right]

X = \left[\begin{array}{cc}-\frac{20}{11} & \frac{2}{11} \\ \frac{57}{22} & \frac{-61}{22}\end{array}\right]

Conclusion

In this article, we solved for the matrix X in the equation $\left[\begin{array}{cc}2 & -4 \\ 5 & 1\end{array}\right] X = \left[\begin{array}{cc}-14 & -14 \\ -13 & 9\end{array}\right]$ . We used the method of matrix inversion to find the inverse of the matrix A, and then multiplied the inverse of A by the matrix B to find the matrix X.

The final answer is:

X = \left[\begin{array}{cc}-\frac{20}{11} & \frac{2}{11} \\ \frac{57}{22} & \frac{-61}{22}\end{array}\right]$$<br/> # **Solving for the Matrix X in a Linear Equation: Q&A**

Introduction

In our previous article, we solved for the matrix X in the equation $\left[\begin{array}{cc}2 & -4 \\ 5 & 1\end{array}\right] X = \left[\begin{array}{cc}-14 & -14 \\ -13 & 9\end{array}\right]$ . In this article, we will answer some frequently asked questions related to solving for the matrix X in a linear equation.

Q: What is the method of matrix inversion?

A: The method of matrix inversion is a technique used to find the inverse of a matrix, which is denoted by $A^{-1}$ . The inverse of a matrix A is a matrix that, when multiplied by A, results in the identity matrix I.

Q: How do I find the determinant of a matrix?

A: The determinant of a 2x2 matrix A is given by:

\det(A) = ad - bc </span></p> <p>where a, b, c, and d are the elements of the matrix A.</p> <h2><strong>Q: What is the adjugate matrix of a matrix?</strong></h2> <p>A: The adjugate matrix of A is a matrix that is obtained by taking the transpose of the matrix of cofactors of A.</p> <h2><strong>Q: How do I find the inverse of a matrix?</strong></h2> <p>A: To find the inverse of a matrix A, you can use the following formula:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msup><mi>A</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>=</mo><mfrac><mn>1</mn><mrow><mi>det</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>×</mo><mtext>adj</mtext><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A^{-1} = \frac{1}{\det(A)} \times \text{adj}(A) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8641em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.2574em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mop">det</span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">adj</span></span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mclose">)</span></span></span></span></span></p> <p>where <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>det</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\det(A)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">det</span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mclose">)</span></span></span></span> is the determinant of the matrix A, and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>adj</mtext><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\text{adj}(A)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">adj</span></span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mclose">)</span></span></span></span> is the adjugate matrix of A.</p> <h2><strong>Q: What is the identity matrix?</strong></h2> <p>A: The identity matrix I is a square matrix with ones on the main diagonal and zeros elsewhere.</p> <h2><strong>Q: How do I multiply two matrices?</strong></h2> <p>A: To multiply two matrices A and B, you need to follow these steps:</p> <ol> <li>Take the first row of matrix A and multiply it by the first column of matrix B.</li> <li>Take the first row of matrix A and multiply it by the second column of matrix B.</li> <li>Take the second row of matrix A and multiply it by the first column of matrix B.</li> <li>Take the second row of matrix A and multiply it by the second column of matrix B.</li> </ol> <h2><strong>Q: What is the matrix X in the equation <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mo fence="true">[</mo><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>2</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mn>4</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>5</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mi>X</mi><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mn>14</mn></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mn>14</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mn>13</mn></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>9</mn></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">\left[\begin{array}{cc}2 &amp; -4 \\ 5 &amp; 1\end{array}\right] X = \left[\begin{array}{cc}-14 &amp; -14 \\ -13 &amp; 9\end{array}\right]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">5</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord">4</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">]</span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord">14</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord">13</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord">14</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">9</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">]</span></span></span></span></span></span>?</strong></h2> <p>A: The matrix X is given by:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>X</mi><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mfrac><mn>20</mn><mn>11</mn></mfrac></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mfrac><mn>2</mn><mn>11</mn></mfrac></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mfrac><mn>57</mn><mn>22</mn></mfrac></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mfrac><mrow><mo>−</mo><mn>61</mn></mrow><mn>22</mn></mfrac></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">X = \left[\begin{array}{cc}-\frac{20}{11} &amp; \frac{2}{11} \\ \frac{57}{22} &amp; \frac{-61}{22}\end{array}\right] </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4102em;vertical-align:-0.9551em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4551em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">11</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">20</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.4049em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">22</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">57</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.9551em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4551em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">11</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.4049em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">22</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">61</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.9551em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">]</span></span></span></span></span></span></span></p> <h2><strong>Q: How do I use the method of matrix inversion to solve for the matrix X in a linear equation?</strong></h2> <p>A: To use the method of matrix inversion to solve for the matrix X in a linear equation, you need to follow these steps:</p> <ol> <li>Find the determinant of the matrix A.</li> <li>Find the adjugate matrix of A.</li> <li>Find the inverse of A using the formula <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>A</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>=</mo><mfrac><mn>1</mn><mrow><mi>det</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>×</mo><mtext>adj</mtext><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A^{-1} = \frac{1}{\det(A)} \times \text{adj}(A)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.3651em;vertical-align:-0.52em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mop mtight"><span class="mtight">d</span><span class="mtight">e</span><span class="mtight">t</span></span><span class="mopen mtight">(</span><span class="mord mathnormal mtight">A</span><span class="mclose mtight">)</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.52em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">adj</span></span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mclose">)</span></span></span></span>.</li> <li>Multiply the inverse of A by the matrix B to find the matrix X.</li> </ol> <h2><strong>Conclusion</strong></h2> <p>In this article, we answered some frequently asked questions related to solving for the matrix X in a linear equation. We also provided a step-by-step guide on how to use the method of matrix inversion to solve for the matrix X in a linear equation.</p> <h2><strong>Frequently Asked Questions</strong></h2> <ul> <li>Q: What is the method of matrix inversion?</li> <li>A: The method of matrix inversion is a technique used to find the inverse of a matrix, which is denoted by <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>A</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">A^{-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span>.</li> <li>Q: How do I find the determinant of a matrix?</li> <li>A: The determinant of a 2x2 matrix A is given by <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>det</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><mi>a</mi><mi>d</mi><mo>−</mo><mi>b</mi><mi>c</mi></mrow><annotation encoding="application/x-tex">\det(A) = ad - bc</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">det</span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">a</span><span class="mord mathnormal">d</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">b</span><span class="mord mathnormal">c</span></span></span></span>.</li> <li>Q: What is the adjugate matrix of a matrix?</li> <li>A: The adjugate matrix of A is a matrix that is obtained by taking the transpose of the matrix of cofactors of A.</li> <li>Q: How do I find the inverse of a matrix?</li> <li>A: To find the inverse of a matrix A, you can use the formula <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>A</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>=</mo><mfrac><mn>1</mn><mrow><mi>det</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>×</mo><mtext>adj</mtext><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A^{-1} = \frac{1}{\det(A)} \times \text{adj}(A)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.3651em;vertical-align:-0.52em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mop mtight"><span class="mtight">d</span><span class="mtight">e</span><span class="mtight">t</span></span><span class="mopen mtight">(</span><span class="mord mathnormal mtight">A</span><span class="mclose mtight">)</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.52em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">adj</span></span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mclose">)</span></span></span></span>.</li> <li>Q: What is the identity matrix?</li> <li>A: The identity matrix I is a square matrix with ones on the main diagonal and zeros elsewhere.</li> <li>Q: How do I multiply two matrices?</li> <li>A: To multiply two matrices A and B, you need to follow these steps: 1. Take the first row of matrix A and multiply it by the first column of matrix B. 2. Take the first row of matrix A and multiply it by the second column of matrix B. 3. Take the second row of matrix A and multiply it by the first column of matrix B. 4. Take the second row of matrix A and multiply it by the second column of matrix B.</li> <li>Q: What is the matrix X in the equation <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mo fence="true">[</mo><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>2</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mn>4</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>5</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mi>X</mi><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mn>14</mn></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mn>14</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mn>13</mn></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>9</mn></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">\left[\begin{array}{cc}2 &amp; -4 \\ 5 &amp; 1\end{array}\right] X = \left[\begin{array}{cc}-14 &amp; -14 \\ -13 &amp; 9\end{array}\right]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">5</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord">4</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">]</span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord">14</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord">13</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord">14</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">9</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">]</span></span></span></span></span></span>?</li> <li>A: The matrix X is given by <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mfrac><mn>20</mn><mn>11</mn></mfrac></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mfrac><mn>2</mn><mn>11</mn></mfrac></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mfrac><mn>57</mn><mn>22</mn></mfrac></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mfrac><mrow><mo>−</mo><mn>61</mn></mrow><mn>22</mn></mfrac></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">X = \left[\begin{array}{cc}-\frac{20}{11} &amp; \frac{2}{11} \\ \frac{57}{22} &amp; \frac{-61}{22}\end{array}\right]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4102em;vertical-align:-0.9551em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4551em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">11</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">20</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.4049em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">22</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">57</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.9551em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4551em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">11</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.4049em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">22</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">61</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.9551em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">]</span></span></span></span></span></span>.</li> <li>Q: How do I use the method of matrix inversion to solve for the matrix X in a linear equation?</li> <li>A: To use the method of matrix inversion to solve for the matrix X in a linear equation, you need to follow these steps: 1. Find the determinant of the matrix A. 2. Find the adjugate matrix of A. 3. Find the inverse of A using the formula <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>A</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>=</mo><mfrac><mn>1</mn><mrow><mi>det</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>×</mo><mtext>adj</mtext><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A^{-1} = \frac{1}{\det(A)} \times \text{adj}(A)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.3651em;vertical-align:-0.52em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mop mtight"><span class="mtight">d</span><span class="mtight">e</span><span class="mtight">t</span></span><span class="mopen mtight">(</span><span class="mord mathnormal mtight">A</span><span class="mclose mtight">)</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.52em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">adj</span></span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mclose">)</span></span></span></span>. 4. Multiply the inverse of A by the matrix B to find the matrix X.</li> </ul>