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

Verified step by step guidance

1. Step 1: Write down the matrices A and B clearly: \(A = \begin{bmatrix} -4 & 0 \\ 1 & 3 \end{bmatrix}\) and \(B = \begin{bmatrix} -2 & 4 \\ 0 & 1 \end{bmatrix}\)
2. Step 2: Calculate the product AB by multiplying 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: \( (AB)_{ij} = \sum_{k=1}^{n} A_{ik} \times B_{kj} \)
3. Step 3: Calculate each element of AB: - For element (1,1): multiply the first row of A by the first column of B - For element (1,2): multiply the first row of A by the second column of B - For element (2,1): multiply the second row of A by the first column of B - For element (2,2): multiply the second row of A by the second column of B
4. Step 4: Calculate the product BA by multiplying matrix B by matrix A using the same method as in Step 2, but switching the order of multiplication.
5. Step 5: Compare the products AB and BA to the identity matrix \(I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\). If both AB and BA equal the identity matrix, then B is the multiplicative inverse of 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 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. For a 2x2 matrix, it is [[1, 0], [0, 1]]. Verifying if AB and BA equal the identity matrix confirms if B is the inverse of A.