Find k
Given:
\[ \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \begin{bmatrix} 3 & 1 \\ 2 & 5 \end{bmatrix} = \begin{bmatrix} 7 & 11 \\ k & 23 \end{bmatrix} \]
Step 1: Multiply Matrices
\[ AB = \begin{bmatrix} 1(3) + 2(2) & 1(1) + 2(5) \\ 3(3) + 4(2) & 3(1) + 4(5) \end{bmatrix} \]
\[ AB = \begin{bmatrix} 3 + 4 & 1 + 10 \\ 9 + 8 & 3 + 20 \end{bmatrix} = \begin{bmatrix} 7 & 11 \\ 17 & 23 \end{bmatrix} \]
Step 2: Compare
\[ k = 17 \]
Final Answer:
\[ \boxed{k = 17} \]