Determining the Nature of the Solution

Oct 13, 2025 by miodeya.com 62 views

Introduction to Systems of Equations

A system of equations is a set of multiple equations that involve the same variables. In this case, we have a matrix representing a system of three linear equations with four variables. The matrix is given as:

{ \begin{bmatrix} 1 & 1 & -2 & | & 1 \\ 2 & -3 & 1 & | & -2 \\ 2 & 2 & -4 & | & 2 \end{bmatrix} \}

Our goal is to determine the nature of the solution to this system of equations. We will explore the different possibilities and use various methods to analyze the matrix and find the solution.

Understanding the Matrix

The matrix represents a system of three linear equations with four variables. The variables are represented by the columns of the matrix, and the equations are represented by the rows. The vertical line | separates the coefficients of the variables from the constants on the right-hand side.

To analyze the matrix, we need to understand the concept of linear independence and dependence. If the columns of the matrix are linearly independent, then the system of equations has a unique solution. If the columns are linearly dependent, then the system has either no solution or infinitely many solutions.

Determining the Nature of the Solution

To determine the nature of the solution, we can use various methods such as Gaussian elimination, row reduction, or the determinant method. In this case, we will use the determinant method to analyze the matrix.

The determinant of a 3x3 matrix is given by:

{ \begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} = a(ei - fh) - b(di - fg) + c(dh - eg) \}

We can calculate the determinant of the given matrix using this formula.

Calculating the Determinant

Let's calculate the determinant of the given matrix:

{ \begin{vmatrix} 1 & 1 & -2 \\ 2 & -3 & 1 \\ 2 & 2 & -4 \end{vmatrix} = 1((-3)(-4) - (1)(2)) - 1((2)(-4) - (1)(2)) + (-2)((2)(2) - (-3)(2)) \}

Simplifying the expression, we get:

{ = 1(12 - 2) - 1(-8 - 2) + (-2)(4 + 6) \}

{ = 1(10) - 1(-10) + (-2)(10) \}

{ = 10 + 10 - 20 \}

{ = 0 \}

The determinant of the matrix is zero. This means that the columns of the matrix are linearly dependent.

Conclusion

Since the determinant of the matrix is zero, the columns of the matrix are linearly dependent. This means that the system of equations has either no solution or infinitely many solutions.

To determine the exact nature of the solution, we need to perform further analysis. We can use Gaussian elimination or row reduction to transform the matrix into row echelon form.

Row Echelon Form

Let's perform row reduction on the matrix to transform it into row echelon form:

{ \begin{bmatrix} 1 & 1 & -2 & | & 1 \\ 2 & -3 & 1 & | & -2 \\ 2 & 2 & -4 & | & 2 \end{bmatrix} \}

Performing row operations, we get:

{ \begin{bmatrix} 1 & 1 & -2 & | & 1 \\ 0 & -5 & 5 & | & -4 \\ 0 & 0 & 0 & | & 0 \end{bmatrix} \}

The matrix is now in row echelon form. We can see that the third row is a zero row, which means that the system of equations has infinitely many solutions.

Infinitely Many Solutions

Since the third row is a zero row, we can conclude that the system of equations has infinitely many solutions. The variables are free to take on any values that satisfy the first two equations.

Conclusion

In conclusion, the matrix represents a system of equations with infinitely many solutions. The determinant of the matrix is zero, which means that the columns of the matrix are linearly dependent. Performing row reduction, we transformed the matrix into row echelon form and found that the third row is a zero row, indicating that the system has infinitely many solutions.

Final Answer

The final answer is: B