In Exercises 1 - 12, find the products AB and BA to determine whether B is the multiplicative inverse of A. 4 - 3 4 3 A = B = - 5 4 5 4

Verified step by step guidance

1. Step 1: Write down the matrices A and B clearly. Matrix A is \( A = \begin{bmatrix} 4 & -3 \\ -5 & 4 \end{bmatrix} \) and matrix B is \( B = \begin{bmatrix} 4 & 3 \\ 5 & 4 \end{bmatrix} \).
2. Step 2: To find the product \( AB \), multiply matrix A by matrix B. Recall that the element in the ith row and jth column of the product is found by taking the dot product of the ith row of A with the jth column of B. For example, the element in the first row and first column of \( AB \) is \( (4)(4) + (-3)(5) \).
3. Step 3: Similarly, calculate all four elements of the product \( AB \) by performing the dot products for each position: first row first column, first row second column, second row first column, and second row second column.
4. Step 4: Next, find the product \( BA \) by multiplying matrix B by matrix A using the same method: the element in the ith row and jth column of \( BA \) is the dot product of the ith row of B with the jth column of A.
5. Step 5: After computing both \( AB \) and \( BA \), compare each product to the identity matrix \( I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \). If both \( AB = I \) and \( BA = I \), then matrix B is the multiplicative inverse of matrix A.

Key Concepts

Matrix Multiplication

Matrix multiplication involves multiplying rows of the first matrix by columns of the second matrix and summing the products. It is essential to compute the products AB and BA correctly to verify if B is the inverse of A. Note that matrix multiplication is not commutative, so AB and BA may differ.

Multiplicative Inverse of a Matrix

A matrix B is the multiplicative inverse of matrix A if both AB and BA equal the identity matrix. The identity matrix acts like 1 in matrix multiplication, having 1s on the diagonal and 0s elsewhere. Finding the inverse is crucial for solving matrix equations and understanding linear transformations.

Identity Matrix

The identity matrix is a square matrix with ones on the main diagonal and zeros elsewhere. It serves as the multiplicative identity in matrix algebra, meaning any matrix multiplied by the identity matrix remains unchanged. Verifying if AB and BA equal the identity matrix confirms if B is the inverse of A.