Determine if \( \left[\begin{array}{rr}-4 & -5 \\ -3 & -4\end{array}\right] \) and \( \left[\begin{array}{rr}-4 & 5 \\ 3 & -4\end{array}\right] \) are inverses of each other.

To determine if the two matrices \[ A = \left[\begin{array}{rr}-4 & -5 \\ -3 & -4\end{array}\right] \] and \[ B = \left[\begin{array}{rr}-4 & 5 \\ 3 & -4\end{array}\right] \] are inverses of each other, we need to check if the product \( AB \) equals the identity matrix \( I \). The identity matrix \( I \) for \( 2 \times 2 \) matrices is \[ I = \left[\begin{array}{rr}1 & 0 \\ 0 & 1\end{array}\right]. \] Now, we will calculate the product \( AB \): \[ AB = \left[\begin{array}{rr}-4 & -5 \\ -3 & -4\end{array}\right] \left[\begin{array}{rr}-4 & 5 \\ 3 & -4\end{array}\right]. \] To compute this product, we use the formula for matrix multiplication: \[ AB = \left[\begin{array}{cc} a_{11}b_{11} + a_{12}b_{21} & a_{11}b_{12} + a_{12}b_{22} \\ a_{21}b_{11} + a_{22}b_{21} & a_{21}b_{12} + a_{22}b_{22} \end{array}\right]. \] Substituting the values from matrices \( A \) and \( B \): 1. First element (row 1, column 1): \[ -4 \cdot -4 + -5 \cdot 3 = 16 - 15 = 1. \] 2. Second element (row 1, column 2): \[ -4 \cdot 5 + -5 \cdot -4 = -20 + 20 = 0. \] 3. Third element (row 2, column 1): \[ -3 \cdot -4 + -4 \cdot 3 = 12 - 12 = 0. \] 4. Fourth element (row 2, column 2): \[ -3 \cdot 5 + -4 \cdot -4 = -15 + 16 = 1. \] Putting it all together, we have: \[ AB = \left[\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}\right] = I. \] Since \( AB = I \), we conclude that the matrices \( A \) and \( B \) are indeed inverses of each other. Thus, the answer is: Yes, the matrices are inverses of each other.