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

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 \(BA\) to check 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 \(I\) of the same size.
2. Step 2: Write down the matrices explicitly: \(A = \begin{bmatrix} 1 & 2 & 3 \\ 1 & 3 & 4 \\ 1 & 4 & 3 \end{bmatrix}\) \(B = \begin{bmatrix} \frac{7}{2} & -3 & \frac{1}{2} \\ -\frac{1}{2} & 0 & \frac{1}{2} \\ -\frac{1}{2} & 1 & -\frac{1}{2} \end{bmatrix}\)
3. Step 3: Calculate the product \(AB\). To do this, multiply each row of \(A\) by each column of \(B\) and sum the products. For example, the element in the first row and first column of \(AB\) is: \( (1)(\frac{7}{2}) + (2)(-\frac{1}{2}) + (3)(-\frac{1}{2}) \). Repeat this for all elements to form the matrix \(AB\).
4. Step 4: Calculate the product \(BA\) similarly by multiplying each row of \(B\) by each column of \(A\). For example, the element in the first row and first column of \(BA\) is: \( (\frac{7}{2})(1) + (-3)(1) + (\frac{1}{2})(1) \). Complete this for all elements to form the matrix \(BA\).
5. Step 5: Compare the resulting matrices \(AB\) and \(BA\) to the identity matrix \(I = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}\). If both \(AB\) and \(BA\) equal \(I\), 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 products AB and BA equal the identity matrix. The identity matrix has 1s on the diagonal and 0s elsewhere. Only square matrices with nonzero 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.