Given A = $\begin{bmatrix$}[r] p & 0 \ 0 & 2 \end{bmatrix}, B = \begin{bmatrix}[r] 0 & -q \ 1 & 0 \end{bmatrix}, C = \begin{bmatrix}[r] 2 & -2 \ 2 & 2 \end{bmatrix} and BA = C². Find the values of p and q.

Given,

$\Rightarrow BA = C^2 \\[1em] \Rightarrow \begin{bmatrix$}[r] 0 & -q \ 1 & 0 \end{bmatrix}\begin{bmatrix}[r] p & 0 \ 0 & 2 \end{bmatrix} = \begin{bmatrix}[r] 2 & -2 \ 2 & 2 \end{bmatrix}\begin{bmatrix}[r] 2 & -2 \ 2 & 2 \end{bmatrix} \\[1em] \Rightarrow \begin{bmatrix}[r] 0 \times p + (-q) \times 0 & 0 \times 0 + (-q) \times 2 \ 1 \times p + 0 \times 0 & 1 \times 0 + 0 \times 2 \end{bmatrix} = \begin{bmatrix}[r] 2 \times 2 + (-2) \times 2 & 2 \times (-2) + (-2) \times 2 \ 2 \times 2 + 2 \times 2 & 2 \times (-2) + 2 \times 2 \end{bmatrix} \\[1em] \Rightarrow \begin{bmatrix}[r] 0 & -2q \ p & 0 \end{bmatrix} = \begin{bmatrix}[r] 0 & -8 \ 8 & 0 \end{bmatrix}

By definition of equality of matrices we get,

-2q = -8
⇒ q = 4.

p = 8.

Hence, p = 8 and q = 4.