Difference Between A and B

Oct 13, 2025 by miodeya.com 117 views

Introduction

In this problem, we are given two matrices A and B, and we need to find the value of A - B. To do this, we first need to find the inverse matrix A and the transpose matrix B. We will then use these matrices to calculate the difference between A and B.

Step 1: Find the Inverse Matrix A

The inverse matrix A is denoted by A^-1. To find the inverse matrix A, we need to follow these steps:

Find the determinant of matrix A.
Find the cofactor matrix of A.
Find the adjugate matrix of A.
Divide the adjugate matrix by the determinant of A.

Step 2: Find the Transpose Matrix B

The transpose matrix B is denoted by B'. To find the transpose matrix B, we need to swap the rows and columns of matrix B.

Step 3: Calculate the Difference Between A and B

Now that we have the inverse matrix A and the transpose matrix B, we can calculate the difference between A and B using the following formula:

A - B = A^-1 - B'

Calculations

Let's start by finding the inverse matrix A.

Matrix A

Matrix A is given by:

A = (1 -2) 3 -4

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

det(A) = 1(3) - (-2)(-4) = 3 - 8 = -5

Cofactor Matrix of A

The cofactor matrix of A is given by:

C = (-1)^(i+j) M_ij

where M_ij is the minor matrix obtained by removing the ith row and jth column of A.

C = (-1)^(1+1) (3) (-1)^(1+2) (-4) (-1)^(2+1) (-2) (-1)^(2+2) (1) = (3) (-4) (2) (1)

Adjugate Matrix of A

The adjugate matrix of A is given by:

adj(A) = C^T

adj(A) = (3) (-4) (2) (1)

Inverse Matrix A

The inverse matrix A is given by:

A^-1 = adj(A) / det(A)

A^-1 = (3) (-4) (2) (1) / (-5) = (-3/5) (4/5) (-2/5) (-1/5)

Matrix B

Matrix B is given by:

B = (A+B -³/²) 1 A

To find the transpose matrix B, we need to swap the rows and columns of matrix B.

B' = (A+B -³/²)' 1 A'

Transpose Matrix B

The transpose matrix B is given by:

B' = (A+B -³/²)' 1 A'

Difference Between A and B

Now that we have the inverse matrix A and the transpose matrix B, we can calculate the difference between A and B using the following formula:

A - B = A^-1 - B'

A - B = (-3/5) (4/5) (-2/5) (-1/5) - (A+B -³/²)' 1 A'

Simplifying the Expression

To simplify the expression, we can start by finding the value of (A+B -³/²).

(A+B -³/²) = (1 -2) 3 -4 + (A+B -³/²)

(A+B -³/²) = (1 -2) 3 -4 + (1 -2) 3 -4

Q&A

Q: What is the inverse matrix A?

A: The inverse matrix A is denoted by A^-1. To find the inverse matrix A, we need to follow these steps:

Find the determinant of matrix A.
Find the cofactor matrix of A.
Find the adjugate matrix of A.
Divide the adjugate matrix by the determinant of A.

Q: What is the transpose matrix B?

A: The transpose matrix B is denoted by B'. To find the transpose matrix B, we need to swap the rows and columns of matrix B.

Q: How do we calculate the difference between A and B?

A: To calculate the difference between A and B, we use the following formula:

A - B = A^-1 - B'

Q: What is the value of (A+B -³/²)?

A: To find the value of (A+B -³/²), we need to add matrix A and matrix B and then divide the result by 2.

Q: How do we simplify the expression A - B?

A: To simplify the expression A - B, we can start by finding the value of (A+B -³/²) and then substitute this value into the expression.

Q: What is the final answer?

A: After simplifying the expression A - B, we get:

A - B = (-3/5) (4/5) (-2/5) (-1/5) - (A+B -³/²)' 1 A'

Since we are given that A ≠ B, we can conclude that the value of A - B is not equal to 0.

Conclusion

In this problem, we were given two matrices A and B, and we needed to find the value of A - B. We used the formula A - B = A^-1 - B' to calculate the difference between A and B. After simplifying the expression, we found that the value of A - B is not equal to 0.

Final Answer

The final answer is E. -3.