Matrix Addition and Subtraction

Oct 13, 2025 by miodeya.com 126 views

Introduction

In linear algebra, matrices are a fundamental concept used to represent systems of equations, linear transformations, and more. When working with matrices, it's essential to understand various operations, such as addition, subtraction, multiplication, and more. In this article, we'll explore matrix operations and help you select the correct answer for a given problem.

Matrix Addition and Subtraction

Matrix addition and subtraction involve adding or subtracting corresponding elements of two matrices. For matrix addition, the resulting matrix has elements that are the sum of the corresponding elements of the two matrices. For matrix subtraction, the resulting matrix has elements that are the difference of the corresponding elements of the two matrices.

Example 1: Matrix Addition

Consider two matrices A and B:

A = \begin{bmatrix} 3 & -7 \\ 6 & 1 \\ -8 & 5 \end{bmatrix}, \quad B = \begin{bmatrix} 4 & -1 \\ 0 & 8 \\ -5 & 3 \end{bmatrix}

To add matrices A and B, we add corresponding elements:

A + B = \begin{bmatrix} 3+4 & -7-1 \\ 6+0 & 1+8 \\ -8-5 & 5+3 \end{bmatrix} = \begin{bmatrix} 7 & -8 \\ 6 & 9 \\ -13 & 8 \end{bmatrix}

Example 2: Matrix Subtraction

To subtract matrix B from matrix A, we subtract corresponding elements:

A - B = \begin{bmatrix} 3-4 & -7+1 \\ 6-0 & 1-8 \\ -8+5 & 5-3 \end{bmatrix} = \begin{bmatrix} -1 & -6 \\ 6 & -7 \\ -3 & 2 \end{bmatrix}

Matrix Multiplication

Matrix multiplication involves multiplying corresponding elements of two matrices and summing the products. The resulting matrix has elements that are the dot product of rows of the first matrix and columns of the second matrix.

Example 1: Matrix Multiplication

Consider two matrices A and B:

A = \begin{bmatrix} 3 & -7 \\ 6 & 1 \\ -8 & 5 \end{bmatrix}, \quad B = \begin{bmatrix} 4 & -1 \\ 0 & 8 \\ -5 & 3 \end{bmatrix}

To multiply matrices A and B, we multiply corresponding elements and sum the products:

AB = \begin{bmatrix} (3)(4)+(-7)(0) & (3)(-1)+(-7)(8) \\ (6)(4)+(1)(0) & (6)(-1)+(1)(8) \\ (-8)(4)+(5)(0) & (-8)(-1)+(5)(8) \end{bmatrix} = \begin{bmatrix} 12 & -61 \\ 24 & -2 \\ -32 & 68 \end{bmatrix}

Which Matrix Represents the Given Operation?

Now that we've explored matrix operations, let's consider the given problem:

Which matrix represents the operation of adding matrix A and matrix B?

A) $ A + B = \begin{bmatrix} 7 & -8 \ 6 & 9 \ -13 & 8 \end{bmatrix} $ B) $ A - B = \begin{bmatrix} -1 & -6 \ 6 & -7 \ -3 & 2 \end{bmatrix} $ C) $ AB = \begin{bmatrix} 12 & -61 \ 24 & -2 \ -32 & 68 \end{bmatrix} $ D) $ BA = \begin{bmatrix} 12 & -61 \ 24 & -2 \ -32 & 68 \end{bmatrix} $

Answer

The correct answer is A) $ A + B = \begin{bmatrix} 7 & -8 \ 6 & 9 \ -13 & 8 \end{bmatrix} $. This matrix represents the operation of adding matrix A and matrix B.

Conclusion

In this article, we explored matrix operations, including addition, subtraction, and multiplication. We also considered a problem that asked us to select the correct matrix that represents a given operation. By understanding matrix operations, we can solve problems and make informed decisions in various fields, such as science, engineering, and finance.

References

[1] Linear Algebra and Its Applications, 4th Edition, by Gilbert Strang
[2] Matrix Algebra, 2nd Edition, by James E. Gentle
[3] Introduction to Linear Algebra, 5th Edition, by Gilbert Strang

Introduction

In our previous article, we explored matrix operations, including addition, subtraction, and multiplication. In this article, we'll answer some frequently asked questions about matrix operations.

Q: What is the difference between matrix addition and matrix multiplication?

A: Matrix addition involves adding corresponding elements of two matrices, while matrix multiplication involves multiplying corresponding elements of two matrices and summing the products.

Q: Can I add or multiply matrices of different sizes?

A: No, matrices must have the same number of rows and columns to be added or multiplied. If the matrices have different sizes, they cannot be added or multiplied.

Q: What is the identity matrix?

A: The identity matrix is a square matrix with ones on the main diagonal and zeros elsewhere. It is used as a multiplicative identity in matrix multiplication.

Q: Can I multiply two matrices that are not square?

A: No, matrices must be square to be multiplied. If the matrices are not square, they cannot be multiplied.

Q: What is the transpose of a matrix?

A: The transpose of a matrix is obtained by interchanging its rows and columns.

Q: Can I add or multiply a matrix with a scalar?

A: Yes, you can add or multiply a matrix with a scalar. This is known as scalar multiplication.

Q: What is the determinant of a matrix?

A: The determinant of a matrix is a scalar value that can be used to determine the invertibility of the matrix.

Q: Can I invert a matrix that is not square?

A: No, matrices must be square to be inverted. If the matrices are not square, they cannot be inverted.

Q: What is the inverse of a matrix?

A: The inverse of a matrix is a matrix that, when multiplied by the original matrix, results in the identity matrix.

Q: Can I use matrix operations to solve systems of equations?

A: Yes, matrix operations can be used to solve systems of equations. This is known as Gaussian elimination.

Q: What are some common applications of matrix operations?

A: Matrix operations have many applications in science, engineering, and finance, including:

Linear algebra
Calculus
Differential equations
Statistics
Machine learning
Computer graphics

Conclusion

In this article, we answered some frequently asked questions about matrix operations. We hope this article has been helpful in clarifying some common concepts and applications of matrix operations.

References

[1] Linear Algebra and Its Applications, 4th Edition, by Gilbert Strang
[2] Matrix Algebra, 2nd Edition, by James E. Gentle
[3] Introduction to Linear Algebra, 5th Edition, by Gilbert Strang