1. The problem is to solve the matrix equation: $$\begin{pmatrix}1 & -2 \\ 3 & -1\end{pmatrix} \begin{pmatrix}2 & 3 \\ 6 & 5\end{pmatrix} = \begin{pmatrix}3 & 2 \\ 1 & -2\end{pmatrix}$$ 2. First, multiply the two matrices on the left side using the rule for matrix multiplication: the element in row $i$, column $j$ of the product is the sum of products of elements from row $i$ of the first matrix and column $j$ of the second matrix. 3. Calculate each element of the product matrix: - Element (1,1): $1 \times 2 + (-2) \times 6 = 2 - 12 = -10$ - Element (1,2): $1 \times 3 + (-2) \times 5 = 3 - 10 = -7$ - Element (2,1): $3 \times 2 + (-1) \times 6 = 6 - 6 = 0$ - Element (2,2): $3 \times 3 + (-1) \times 5 = 9 - 5 = 4$ 4. So the product matrix is: $$\begin{pmatrix}-10 & -7 \\ 0 & 4\end{pmatrix}$$ 5. The problem states this product equals: $$\begin{pmatrix}3 & 2 \\ 1 & -2\end{pmatrix}$$ 6. Since the product matrix does not equal the given matrix, the equation as stated is false. If the problem is to find unknowns, there are none here, so the equation is inconsistent. 7. Therefore, the matrix multiplication does not equal the given matrix, and no solution exists for this equation as is. Final answer: $$\begin{pmatrix}1 & -2 \\ 3 & -1\end{pmatrix} \begin{pmatrix}2 & 3 \\ 6 & 5\end{pmatrix} = \begin{pmatrix}-10 & -7 \\ 0 & 4\end{pmatrix} \neq \begin{pmatrix}3 & 2 \\ 1 & -2\end{pmatrix}$$