Question
If \[ \begin{bmatrix} 2 & 3 \\ 5 & 7 \end{bmatrix} \begin{bmatrix} 1 & -3 \\ -2 & 4 \end{bmatrix} = \begin{bmatrix} -4 & 6 \\ -9 & x \end{bmatrix}, \] find \(x\).
Solution
\[ \begin{bmatrix} 2 & 3 \\ 5 & 7 \end{bmatrix} \begin{bmatrix} 1 & -3 \\ -2 & 4 \end{bmatrix} = \begin{bmatrix} 2\cdot1 + 3(-2) & 2(-3) + 3\cdot4 \\ 5\cdot1 + 7(-2) & 5(-3) + 7\cdot4 \end{bmatrix} \] —Simplifying
\[ = \begin{bmatrix} 2 – 6 & -6 + 12 \\ 5 – 14 & -15 + 28 \end{bmatrix} = \begin{bmatrix} -4 & 6 \\ -9 & 13 \end{bmatrix} \] —Compare with given matrix
\[ \begin{bmatrix} -4 & 6 \\ -9 & 13 \end{bmatrix} = \begin{bmatrix} -4 & 6 \\ -9 & x \end{bmatrix} \] \[ x = 13 \] —Final Answer
\[
x = 13
\]
Hence, the required value is \(x = 13\).