Problem 11 - Precalculus with Limits

Question

In Exercises \(5-12,\) show that \(B\) is the inverse of \(A .\) $$A=\left[\begin{array}{rrr}{-2} & {2} & {3} \\ {1} & {-1} & {0} \\ {0} & {1} & {4}\end{array}\right], B=\frac{1}{3}\left[\begin{array}{rrr}{-4} & {-5} & {3} \\ {-4} & {-8} & {3} \\ {1} & {2} & {0}\end{array}\right]$$

Step-by-Step Solution

Answer

By performing the matrix multiplications \(AB\) and \(BA\), and verifying that both results are the identity matrix, we can confirm that \(B\) is the inverse of \(A\).

Step 1: Calculating \(AB\)

To calculate the product \(AB\), perform the matrix multiplication. That means, for each cell in the resulting matrix, sum the products of corresponding elements from the row of the first matrix and column of the second one.

Step 2: Verifying the Result of \(AB\)

Check if the resulting matrix from Step 1 is the identity. If it is, this is a good sign but we still need to multiply in the other direction, because matrix multiplication is not necessarily commutative, so \(AB\) might be different from \(BA\).

Step 3: Calculating \(BA\)

Similar to step 1, now calculate the product \(BA\) by performing the matrix multiplication.

Step 4: Verifying the Result of \(BA\)

Check if the resulting matrix from Step 3 is the identity. If so, that confirms that \(B\) is indeed the inverse of \(A\).