Comparing the Two Matrices

Oct 13, 2025 by windailys.com 75 views

Introduction

Matrix multiplication is a fundamental concept in linear algebra, and it has numerous applications in various fields, including physics, engineering, and computer science. Given two matrices A and B, the product AB is a new matrix that is obtained by multiplying the rows of A with the columns of B. In this article, we will explore how to find the product of two matrices, and we will use the given matrices A and B to illustrate the process.

Matrix A and Matrix B

The given matrices are:

${ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}, \quad B = \begin{bmatrix} 22 & 11 & 7 \\ 6 & -9 & -15 \end{bmatrix} }$

Matrix Multiplication

To find the product AB, we need to multiply the rows of A with the columns of B. The resulting matrix will have the same number of rows as A and the same number of columns as B.

Step 1: Multiply the First Row of A with the First Column of B

To find the first element of the product matrix, we need to multiply the first row of A with the first column of B.

${ \begin{bmatrix} a & b \end{bmatrix} \begin{bmatrix} 22 \\ 6 \end{bmatrix} = a(22) + b(6) = 22a + 6b }$

Step 2: Multiply the First Row of A with the Second Column of B

To find the second element of the product matrix, we need to multiply the first row of A with the second column of B.

${ \begin{bmatrix} a & b \end{bmatrix} \begin{bmatrix} 11 \\ -9 \end{bmatrix} = a(11) + b(-9) = 11a - 9b }$

Step 3: Multiply the First Row of A with the Third Column of B

To find the third element of the product matrix, we need to multiply the first row of A with the third column of B.

${ \begin{bmatrix} a & b \end{bmatrix} \begin{bmatrix} 7 \\ -15 \end{bmatrix} = a(7) + b(-15) = 7a - 15b }$

Step 4: Multiply the Second Row of A with the First Column of B

To find the fourth element of the product matrix, we need to multiply the second row of A with the first column of B.

${ \begin{bmatrix} c & d \end{bmatrix} \begin{bmatrix} 22 \\ 6 \end{bmatrix} = c(22) + d(6) = 22c + 6d }$

Step 5: Multiply the Second Row of A with the Second Column of B

To find the fifth element of the product matrix, we need to multiply the second row of A with the second column of B.

${ \begin{bmatrix} c & d \end{bmatrix} \begin{bmatrix} 11 \\ -9 \end{bmatrix} = c(11) + d(-9) = 11c - 9d }$

Step 6: Multiply the Second Row of A with the Third Column of B

To find the sixth element of the product matrix, we need to multiply the second row of A with the third column of B.

${ \begin{bmatrix} c & d \end{bmatrix} \begin{bmatrix} 7 \\ -15 \end{bmatrix} = c(7) + d(-15) = 7c - 15d }$

The Product Matrix AB

The product matrix AB is:

${ AB = \begin{bmatrix} 22a + 6b & 11a - 9b & 7a - 15b \\ 22c + 6d & 11c - 9d & 7c - 15d \end{bmatrix} }$

Given Product Matrix AB

The given product matrix AB is:

${ AB = \begin{bmatrix} 110 & -77 & -151 \\ 230 & 139 & 107 \end{bmatrix} }$

Comparing the Two Matrices

We can compare the two matrices AB and the given product matrix AB to find the values of a, b, c, and d.

${ 22a + 6b = 110 }$ ${ 11a - 9b = -77 }$ ${ 7a - 15b = -151 }$ ${ 22c + 6d = 230 }$ ${ 11c - 9d = 139 }$ ${ 7c - 15d = 107 }$

Solving the System of Equations

We can solve the system of equations to find the values of a, b, c, and d.

Step 1: Solve the First Two Equations

We can solve the first two equations to find the values of a and b.

${ 22a + 6b = 110 }$ ${ 11a - 9b = -77 }$

Multiplying the first equation by 9 and the second equation by 6, we get:

${ 198a + 54b = 990 }$ ${ 66a - 54b = -462 }$

Adding the two equations, we get:

${ 264a = 528 }$ ${ a = 2 }$

Substituting the value of a into the first equation, we get:

${ 22(2) + 6b = 110 }$ ${ 44 + 6b = 110 }$ ${ 6b = 66 }$ ${ b = 11 }$

Step 2: Solve the Last Three Equations

We can solve the last three equations to find the values of c and d.

${ 22c + 6d = 230 }$ ${ 11c - 9d = 139 }$ ${ 7c - 15d = 107 }$

Multiplying the first equation by 9 and the second equation by 6, we get:

${ 198c + 54d = 2070 }$ ${ 66c - 54d = 834 }$

Adding the two equations, we get:

${ 264c = 2904 }$ ${ c = 11 }$

Substituting the value of c into the first equation, we get:

${ 22(11) + 6d = 230 }$ ${ 242 + 6d = 230 }$ ${ 6d = -12 }$ ${ d = -2 }$

The Values of a, b, c, and d

The values of a, b, c, and d are:

${ a = 2 }$ ${ b = 11 }$ ${ c = 11 }$ ${ d = -2 }$

Conclusion

Introduction

Matrix multiplication is a fundamental concept in linear algebra, and it has numerous applications in various fields, including physics, engineering, and computer science. In our previous article, we explored how to find the product of two matrices, and we used the given matrices A and B to illustrate the process. In this article, we will answer some of the most frequently asked questions about matrix multiplication.

Q: What is matrix multiplication?

A: Matrix multiplication is a mathematical operation that takes two matrices as input and produces another matrix as output. The resulting matrix is obtained by multiplying the rows of the first matrix with the columns of the second matrix.

Q: What are the conditions for matrix multiplication?

A: For matrix multiplication to be possible, the number of columns in the first matrix must be equal to the number of rows in the second matrix. In other words, if A is an m x n matrix and B is an n x p matrix, then the product AB is an m x p matrix.

Q: How do I perform matrix multiplication?

A: To perform matrix multiplication, you need to follow these steps:

Multiply the rows of the first matrix with the columns of the second matrix.
Add the products of the corresponding elements of the rows and columns.
Repeat the process for each element of the resulting matrix.

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

A: Matrix multiplication and matrix addition are two different mathematical operations. Matrix multiplication involves multiplying the rows of one matrix with the columns of another matrix, while matrix addition involves adding corresponding elements of two matrices.

Q: Can I multiply a matrix by a scalar?

A: Yes, you can multiply a matrix by a scalar. This is known as scalar multiplication. To multiply a matrix by a scalar, you need to multiply each element of the matrix by the scalar.

Q: What is the identity matrix?

A: The identity matrix is a special type of matrix that has 1s on the main diagonal and 0s elsewhere. It is used as a multiplicative identity in matrix multiplication.

Q: Can I multiply two identity matrices?

A: Yes, you can multiply two identity matrices. The result is another identity matrix.

Q: What is the inverse of a matrix?

A: The inverse of a matrix is a special type of matrix that, when multiplied by the original matrix, produces the identity matrix.

Q: Can I multiply a matrix by its inverse?

A: Yes, you can multiply a matrix by its inverse. The result is the identity matrix.

Q: What are some common applications of matrix multiplication?

A: Matrix multiplication has numerous applications in various fields, including:

Linear algebra
Physics
Engineering
Computer science
Data analysis

Q: How do I use matrix multiplication in real-world problems?

A: Matrix multiplication is used in a wide range of real-world problems, including:

Image processing
Signal processing
Data analysis
Machine learning
Computer graphics

Conclusion

In this article, we have answered some of the most frequently asked questions about matrix multiplication. We have covered topics such as the conditions for matrix multiplication, how to perform matrix multiplication, and some common applications of matrix multiplication. We hope that this article has been helpful in clarifying any doubts you may have had about matrix multiplication.