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

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 of the same size.
2. Step 2: Write down the matrices explicitly for clarity: A = \begin{bmatrix} 0 & 0 & -2 & 1 \\ -1 & 0 & 1 & 1 \\ 0 & 1 & -1 & 0 \\ 1 & 0 & 0 & -1 \end{bmatrix}, B = \begin{bmatrix} 1 & 2 & 0 & 3 \\ 0 & 1 & 1 & 1 \\ 0 & 1 & 0 & 1 \\ 1 & 2 & 0 & 2 \end{bmatrix}
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 AB, compute the sum of A's i-th row elements multiplied by B's j-th column elements.
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. For each element (i,j) in the product matrix BA, compute the sum of B's i-th row elements multiplied by A's j-th column elements.
5. Step 5: Compare the resulting matrices AB and BA to the 4x4 identity matrix I_4. If both AB = I_4 and BA = I_4, then B is the multiplicative inverse of A. Otherwise, B is not the inverse.

Key Concepts

Matrix Multiplication

Matrix multiplication involves multiplying rows of the first matrix by columns of the second matrix and summing the products. It is essential to understand the dimensions and the order of multiplication, as AB and BA may yield different results or may not even be defined if dimensions do not align.

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 that multiplying A by B returns the identity matrix, which acts like 1 in matrix algebra, confirming that B reverses the effect of A.

Identity Matrix

The identity matrix is a square matrix with ones on the diagonal and zeros elsewhere. It serves as the multiplicative identity in matrix operations, meaning any matrix multiplied by the identity matrix remains unchanged.