In Exercises 1 - 12, find the products AB and BA to determine whether B is the multiplicative inverse of A. - 4 0 - 2 4A = B = 1 3 0 1

Matrix multiplication involves taking the rows of the first matrix and the columns of the second matrix to produce a new matrix. The element in the resulting matrix is calculated by summing the products of corresponding entries. This operation is not commutative, meaning that AB does not necessarily equal BA, which is crucial for determining if B is the multiplicative inverse of A.

A matrix B is considered the multiplicative inverse of matrix A if the product of AB equals the identity matrix I. The identity matrix has 1s on the diagonal and 0s elsewhere. For a 2x2 matrix, this means that the product must yield a matrix that effectively acts as a neutral element in multiplication, confirming that B undoes the effect of A.

The identity matrix is a square matrix with ones on the diagonal and zeros elsewhere. For a 2x2 matrix, it is represented as I = [1 0; 0 1]. When multiplying any matrix by the identity matrix, the original matrix remains unchanged. In the context of this problem, confirming that AB and BA equal the identity matrix is essential to verify that B is indeed the multiplicative inverse of A.