If matrix M = $\begin{bmatrix$}[r] 1 & 1 \ 8 & 3 \end{bmatrix}, find M² - 4M - 4I.

Substituting value of M in M² - 4M - 4I, we get :

$\Rightarrow \begin{bmatrix$}[r] 1 & 1 \ 8 & 3 \end{bmatrix}\begin{bmatrix}[r] 1 & 1 \ 8 & 3 \end{bmatrix} - 4\begin{bmatrix}[r] 1 & 1 \ 8 & 3 \end{bmatrix} - 4\begin{bmatrix}[r] 1 & 0 \ 0 & 1 \end{bmatrix} \\[1em] \Rightarrow \begin{bmatrix}[r] 1 \times 1 + 1 \times 8 & 1 \times 1 + 1 \times 3 \ 8 \times 1 + 3 \times 8 & 8 \times 1 + 3 \times 3 \end{bmatrix} - \begin{bmatrix}[r] 4 & 4 \ 32 & 12 \end{bmatrix} - \begin{bmatrix}[r] 4 & 0 \ 0 & 4 \end{bmatrix} \\[1em] \Rightarrow \begin{bmatrix}[r] 1 + 8 & 1 + 3 \ 8 + 24 & 8 + 9 \end{bmatrix} - \begin{bmatrix}[r] 4 & 4 \ 32 & 12 \end{bmatrix} - \begin{bmatrix}[r] 4 & 0 \ 0 & 4 \end{bmatrix} \\[1em] \Rightarrow \begin{bmatrix}[r] 9 & 4 \ 32 & 17 \end{bmatrix} - \begin{bmatrix}[r] 4 & 4 \ 32 & 12 \end{bmatrix} - \begin{bmatrix}[r] 4 & 0 \ 0 & 4 \end{bmatrix} \\[1em] \Rightarrow \begin{bmatrix}[r] 9 - 4 - 4 & 4 - 4 - 0 \ 32 - 32 - 0 & 17 - 12 - 4 \end{bmatrix} \\[1em] \Rightarrow \begin{bmatrix}[r] 1 & 0 \ 0 & 1 \end{bmatrix}.

Hence, M² - 4M - 4I = $\begin{bmatrix$}[r] 1 & 0 \ 0 & 1 \end{bmatrix}.