In Exercises 1 - 12, find the products AB and BA to determine whether B is the multiplicative inverse of A. 0 1 0 0 0 1A = 0 0 1 B = 1 0 01 0 0 0 1 0

Verified step by step guidance

1. Multiply matrix A by matrix B to find the product AB.
2. For AB, calculate the first row, first column element: (0*0 + 1*1 + 0*0).
3. For AB, calculate the first row, second column element: (0*0 + 1*0 + 0*1).
4. For AB, calculate the first row, third column element: (0*1 + 1*0 + 0*0).
5. Repeat the process for the remaining elements of AB and then calculate BA similarly.

Key Concepts

Matrix Multiplication

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.

Identity Matrix

An identity matrix is a square matrix with ones on the diagonal and zeros elsewhere. It acts as the multiplicative identity in matrix multiplication, meaning that for any matrix A, multiplying by the identity matrix I results in A (AI = A and IA = A). The identity matrix is crucial for determining if a matrix has an inverse.

Multiplicative Inverse

A matrix B is considered the multiplicative inverse of matrix A if the products AB and BA both equal the identity matrix. This means that multiplying A by B (and vice versa) effectively 'cancels out' A, demonstrating that B undoes the transformation represented by A. Finding the inverse is essential in solving systems of equations and other applications.