Given A = $\begin{bmatrix$}[r] 2 & 0 \ -1 & 7 \end{bmatrix} \text{ and } I = \begin{bmatrix}[r] 1 & 0 \ 0 & 1 \end{bmatrix} and A² = 9A + mI. Find m.

Given,

$\Rightarrow A^2 = 9A + mI \\[1em] \Rightarrow \begin{bmatrix$}[r] 2 & 0 \ -1 & 7 \end{bmatrix}\begin{bmatrix}[r] 2 & 0 \ -1 & 7 \end{bmatrix} = 9\begin{bmatrix}[r] 2 & 0 \ -1 & 7 \end{bmatrix} + m\begin{bmatrix}[r] 1 & 0 \ 0 & 1 \end{bmatrix} \\[1em] \Rightarrow \begin{bmatrix}[r] 2 \times 2 + 0 \times (-1) & 2 \times 0 + 0 \times 7 \ -1 \times 2 + 7 \times (-1) & -1 \times 0 + 7 \times 7 \end{bmatrix} = \begin{bmatrix}[r] 18 & 0 \ -9 & 63 \end{bmatrix} + \begin{bmatrix}[r] m & 0 \ 0 & m \end{bmatrix} \\[1em] \Rightarrow \begin{bmatrix}[r] 4 & 0 \ -9 & 49 \end{bmatrix} = \begin{bmatrix}[r] 18 & 0 \ -9 & 63 \end{bmatrix} + \begin{bmatrix}[r] m & 0 \ 0 & m \end{bmatrix} \\[1em] \Rightarrow \begin{bmatrix}[r] m & 0 \ 0 & m \end{bmatrix} = \begin{bmatrix}[r] 4 & 0 \ -9 & 49 \end{bmatrix} - \begin{bmatrix}[r] 18 & 0 \ -9 & 63 \end{bmatrix} \\[1em] \Rightarrow \begin{bmatrix}[r] m & 0 \ 0 & m \end{bmatrix} = \begin{bmatrix}[r] 4 - 18 & 0 - 0 \ -9 - (-9) & 49 - 63 \end{bmatrix} \\[1em] \Rightarrow \begin{bmatrix}[r] m & 0 \ 0 & m \end{bmatrix} = \begin{bmatrix}[r] -14 & 0 \ 0 & -14 \end{bmatrix}

By definition of equality of matrices we get,

⇒ m = -14.

Hence, m = -14.