3. Prove that \( \left[\begin{array}{ll}3 & -1 \\ 5 & -2\end{array}\right] \) and \( \left[\begin{array}{cc}2 & -1 \\ 5 & -3\end{array}\right] \) are the inverses of each oth

To prove that two matrices are inverses of each other, we need to show that their product is the identity matrix. Given matrices: \( A = \left[\begin{array}{ll}3 & -1 \\ 5 & -2\end{array}\right] \) \( B = \left[\begin{array}{cc}2 & -1 \\ 5 & -3\end{array}\right] \) We need to find the product of matrices A and B and check if it is equal to the identity matrix. Let's calculate the product of matrices A and B: \( AB = \left[\begin{array}{ll}3 & -1 \\ 5 & -2\end{array}\right] \times \left[\begin{array}{cc}2 & -1 \\ 5 & -3\end{array}\right] \) \( AB = \left[\begin{array}{ll}(3*2 + (-1)*5) & (3*(-1) + (-1)*(-3)) \\ (5*2 + (-2)*5) & (5*(-1) + (-2)*(-3))\end{array}\right] \) \( AB = \left[\begin{array}{ll}(6 - 5) & (-3 + 3) \\ (10 - 10) & (-5 + 6)\end{array}\right] \) \( AB = \left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right] \) Since the product of matrices A and B is the identity matrix, we have proven that \( \left[\begin{array}{ll}3 & -1 \\ 5 & -2\end{array}\right] \) and \( \left[\begin{array}{cc}2 & -1 \\ 5 & -3\end{array}\right] \) are inverses of each other.