Problem 56 - Algebra and Trigonometry

Question

If you are given two matrices, \(A\) and \(B,\) explain how to determine if \(B\) is the multiplicative inverse of \(A\).

Step-by-Step Solution

Answer

To verify whether a matrix \(B\) is the inverse of \(A\), it is necessary to perform matrix multiplication in both orders, \(A*B\) and \(B*A\) , and see if the result is an identity matrix in both cases.

Step 1: Identify the given Matrices

The first step is to identify the given matrices \(A\) and \(B\). The problem statement usually provides these matrices.

Step 2: Check size of Matrices

For \(B\) to be the inverse of \(A\), both matrices must be square (i.e., the number of rows equals the number of columns). So, both matrices \(A\) and \(B\) must have identical dimensions.

Step 3: Calculate the Product AB

After ensuring the matrices are square and have the same dimensions, the next step is to calculate the product of the two matrices, i.e., \(A*B\).

Step 4: Compare with the Identity Matrix

After performing the matrix multiplication, check if the resulting matrix is an identity matrix. If \(A*B\) gives an identity matrix, then \(B\) is the multiplicative inverse of \(A\).

Step 5: Calculate the Product BA

Matrix multiplication is not commutative, so it is also important to compute \(B*A\).

Step 6: Final Verification Check

If both \(A*B\) and \(B*A\) give the identity matrix \(I\), then the matrices \(A\) and \(B\) are indeed inverses of each other. Otherwise, if either product does not yield the identity matrix, then \(B\) is not the multiplicative inverse of \(A\).