If A = $\begin{bmatrix$}[r] 4 & -4 \ -4 & 4 \end{bmatrix}, find A². If A² = pA, then find the value of p.

Given,

⇒ A² = pA

$\Rightarrow \begin{bmatrix$}[r] 4 & -4 \ -4 & 4 \end{bmatrix}\begin{bmatrix}[r] 4 & -4 \ -4 & 4 \end{bmatrix} = p\begin{bmatrix}[r] 4 & -4 \ -4 & 4 \end{bmatrix} \\[1em] \Rightarrow \begin{bmatrix}[r] 4 \times 4 + (-4) \times (-4) & 4 \times (-4) + (-4) \times 4 \ -4 \times 4 + 4 \times (-4) & (-4) \times (-4) + 4 \times 4 \end{bmatrix} = p\begin{bmatrix}[r] 4 & -4 \ -4 & 4 \end{bmatrix} \\[1em] \Rightarrow \begin{bmatrix}[r] 16 + 16 & -16 - 16 \ -16 - 16 & 16 + 16 \end{bmatrix} = p\begin{bmatrix}[r] 4 & -4 \ -4 & 4 \end{bmatrix} \\[1em] \Rightarrow \begin{bmatrix}[r] 32 & -32 \ -32 & 32 \end{bmatrix} = p\begin{bmatrix}[r] 4 & -4 \ -4 & 4 \end{bmatrix} \\[1em] \Rightarrow 32\begin{bmatrix}[r] 1 & -1 \ -1 & 1 \end{bmatrix} = 4p\begin{bmatrix}[r] 1 & -1 \ -1 & 1 \end{bmatrix} \\[1em] \Rightarrow 32 = 4p \\[1em] \Rightarrow p = \dfrac{32}{4} = 8.

Hence, p = 8.