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
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To prove that the given matrices are inverses of each other, we need to multiply them and check if the result is the identity matrix. Let's denote the first matrix as \( A = \left[\begin{array}{ll}3 & -1 \\ 5 & -2\end{array}\right] \) and the second matrix as \( B = \left[\begin{array}{cc}2 & -1 \\ 5 & -3\end{array}\right] \). Now we calculate the product \( AB \): \[ AB = \left[\begin{array}{ll}3 & -1 \\ 5 & -2\end{array}\right] \left[\begin{array}{cc}2 & -1 \\ 5 & -3\end{array}\right] \] Calculating the elements of the product: 1. First row, first column: \[ (3 \cdot 2) + (-1 \cdot 5) = 6 - 5 = 1 \] 2. First row, second column: \[ (3 \cdot -1) + (-1 \cdot -3) = -3 + 3 = 0 \] 3. Second row, first column: \[ (5 \cdot 2) + (-2 \cdot 5) = 10 - 10 = 0 \] 4. Second row, second column: \[ (5 \cdot -1) + (-2 \cdot -3) = -5 + 6 = 1 \] Putting it all together, we have: \[ AB = \left[\begin{array}{cc}1 & 0 \\ 0 & 1\end{array}\right] \] Since the product \( AB \) is the identity matrix, we conclude that \( A \) and \( B \) are indeed inverses of each other.
