In Exercises 37–38, find the products and to determine whether B is the multiplicative inverse of A.

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Step 1: Recall the definition of the multiplicative inverse. For a matrix B to be the multiplicative inverse of a matrix A, their product must equal the identity matrix. Specifically, A × B = I and B × A = I, where I is the identity matrix of the same dimension as A and B.

Step 2: Multiply matrix A by matrix B. Use the rules of matrix multiplication, which involve taking the dot product of the rows of A with the columns of B. Write out the resulting matrix explicitly.

Step 3: Multiply matrix B by matrix A. Again, use the rules of matrix multiplication, taking the dot product of the rows of B with the columns of A. Write out the resulting matrix explicitly.

Step 4: Compare the results of A × B and B × A to the identity matrix. The identity matrix has 1s along the main diagonal (from top-left to bottom-right) and 0s elsewhere. Check if both products are equal to the identity matrix.

Step 5: Conclude whether B is the multiplicative inverse of A. If both A × B = I and B × A = I, then B is the multiplicative inverse of A. If not, B is not the multiplicative inverse of A.

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Matrix Multiplication

Matrix multiplication involves combining two matrices to produce a third matrix. The number of columns in the first matrix must equal the number of rows in the second matrix. The resulting matrix's elements are calculated by taking the dot product of the rows of the first matrix with the columns of the second matrix. Understanding this operation is crucial for finding the product of matrices A and B.

Multiplicative Inverse

The multiplicative inverse of a matrix A is another matrix, denoted as A⁻¹, such that when A is multiplied by A⁻¹, the result is the identity matrix I. The identity matrix acts like the number 1 in matrix operations, meaning that A * A⁻¹ = I. To determine if B is the multiplicative inverse of A, one must verify if the product AB equals the identity matrix.

Identity Matrix

An identity matrix is a square matrix with ones on the diagonal and zeros elsewhere. It serves as the multiplicative identity in matrix algebra, meaning that any matrix multiplied by the identity matrix remains unchanged. For a matrix A of size n x n, the identity matrix I will also be of size n x n, and confirming that the product of A and its supposed inverse B yields I is essential for validating B as A's inverse.