Question

If

\[ A = \begin{bmatrix} 1 & -1 & 0 \\ 2 & 3 & -2 \\ -2 & 0 & 1 \end{bmatrix}, \quad B = \begin{bmatrix} 3 & -15 & 5 \\ -1 & 6 & -2 \\ 1 & -5 & 2 \end{bmatrix}, \]

then find the value of \( (AB)^{-1} \).

Hint:

The inverse of a product of matrices is the product of their inverses in reverse order: \( (AB)^{-1} = B^{-1} A^{-1} \).

Answer 1

To find \( (AB)^{-1} \), we use the property of inverse matrices:

\[ (AB)^{-1} = B^{-1} A^{-1}. \]

1. Compute \( A^{-1} \): Use the formula \( A^{-1} = \frac{1}{\det(A)} \text{adj}(A) \), where:

\[ \det(A) = \begin{vmatrix} 1 & -1 & 0 \\ 2 & 3 & -2 \\ -2 & 0 & 1 \end{vmatrix}. \]

2. Compute \( B^{-1} \): Similarly, calculate:

\[ B^{-1} = \frac{1}{\det(B)} \text{adj}(B). \]

3. Multiply \( B^{-1} A^{-1} \) to get \( (AB)^{-1} \).

Perform these calculations to get the final result for \( (AB)^{-1} \).