If \(P = \begin{pmatrix} 1 & 2 \\ 5 & 1 \end{pmatrix}\) and \(Q = \begin{pmatrix} 0 & 1 \\ 1 & 3 \end{pmatrix}\), find PQ.

If \(P = \begin{pmatrix} 1 & 2 \\ 5 & 1 \end{pmatrix}\) and \(Q = \begin{pmatrix} 0 & 1 \\ 1 & 3 \end{pmatrix}\), find PQ.

Answer Details

To find the product PQ of matrices P and Q, we multiply the elements of each row of matrix P by the corresponding elements of each column of matrix Q, and add up the products. Thus, we have: \begin{align*} PQ &= \begin{pmatrix} 1 & 2 \\ 5 & 1 \end{pmatrix} \begin{pmatrix} 0 & 1 \\ 1 & 3 \end{pmatrix} \\ &= \begin{pmatrix} (1)(0)+(2)(1) & (1)(1)+(2)(3) \\ (5)(0)+(1)(1) & (5)(1)+(1)(3) \end{pmatrix} \\ &= \begin{pmatrix} 2 & 7 \\ 1 & 8 \end{pmatrix} \end{align*} Therefore, the answer is option (C): \(\begin{pmatrix} 2 & 7 \\ 1 & 8 \end{pmatrix}\).