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 1 A = 0 0 1 B = 1 0 0 1 0 0 0 1 0

Verified step by step guidance

1. Step 1: Understand the problem. You are given two matrices A and B, and you need to find the products AB and BA. Then, determine if B is the multiplicative inverse of A. Recall that if B is the inverse of A, then both AB and BA should equal the identity matrix I.
2. Step 2: Write down the matrices explicitly. Matrix A is $\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 1& 0& 0\end{array}\right]$ and matrix B is $\left[\begin{array}{ccc}0& 0& 1\\ 1& 0& 0\\ 0& 1& 0\end{array}\right]$.
3. Step 3: Calculate the product AB. Multiply matrix A by matrix B by taking the dot product of the rows of A with the columns of B. For each element (i,j) in the product matrix, compute the sum of products of corresponding elements from row i of A and column j of B.
4. Step 4: Calculate the product BA. Similarly, multiply matrix B by matrix A by taking the dot product of the rows of B with the columns of A, following the same procedure as in Step 3.
5. Step 5: Compare the resulting matrices AB and BA to the identity matrix $I=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$. If both AB and BA equal the identity matrix, then B is the multiplicative inverse of A; otherwise, it is not.

Key Concepts

Matrix Multiplication

Matrix multiplication involves multiplying rows of the first matrix by columns of the second matrix and summing the products. The product AB is defined only if the number of columns in A equals the number of rows in B. This operation is not commutative, meaning AB may not equal BA.

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 has 1s on the diagonal and 0s elsewhere. Only square matrices with non-zero determinants have inverses.

Identity Matrix

The identity matrix acts like 1 in matrix multiplication, leaving any matrix unchanged when multiplied. It is a square matrix with 1s on the main diagonal and 0s elsewhere. Verifying if AB and BA equal the identity matrix confirms if B is the inverse of A.