Find the products AB and BA to determine whether B is the multiplicative inverse of A.

Verified step by step guidance

1. Step 1: Identify the matrices A and B. Matrix A is given as \(A = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{bmatrix}\) and matrix B is \(B = \begin{bmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix}\).
2. Step 2: To find the product \(AB\), multiply matrix A by matrix B. Recall that the element in the \(i^{th}\) row and \(j^{th}\) column of the product matrix is found by taking the dot product of the \(i^{th}\) row of A with the \(j^{th}\) column of B. Formally, \((AB)_{ij} = \sum_{k=1}^3 A_{ik} B_{kj}\).
3. Step 3: Perform the multiplication for each element of the product matrix \(AB\) by calculating the dot products row by row and column by column.
4. Step 4: Repeat the process to find the product \(BA\) by multiplying matrix B by matrix A, using the same dot product method for each element: \((BA)_{ij} = \sum_{k=1}^3 B_{ik} A_{kj}\).
5. Step 5: After finding both products \(AB\) and \(BA\), compare each with the identity matrix \(I = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}\). 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. The order of multiplication matters, so AB and BA can yield different results. Understanding this process is essential to compute the products correctly.

Multiplicative Inverse of a Matrix

A matrix B is the multiplicative inverse of matrix A if both AB and BA equal the identity matrix. This means multiplying A by B returns the identity matrix, which acts like 1 in matrix algebra. Verifying both products confirms if B is truly the inverse of A.

Identity Matrix

The identity matrix is a square matrix with 1s on the diagonal and 0s elsewhere. It serves as the multiplicative identity in matrix operations, meaning any matrix multiplied by the identity matrix remains unchanged. Recognizing the identity matrix is key to checking inverses.