If A = $\begin{bmatrix$}[r] 3 & x \ 0 & 1 \end{bmatrix} \text{ and B} = \begin{bmatrix}[r] 9 & 16 \ 0 & -y \end{bmatrix}, find x and y when A² = B.

Given,

A² = B

$\Rightarrow \begin{bmatrix$}[r] 3 & x \ 0 & 1 \end{bmatrix} \begin{bmatrix}[r] 3 & x \ 0 & 1 \end{bmatrix} = \begin{bmatrix}[r] 9 & 16 \ 0 & -y \end{bmatrix} \\[1em] \Rightarrow \begin{bmatrix}[r] 3 \times 3 + x \times 0 & 3 \times x + x \times 1 \ 0 \times 3 + 1 \times 0 & 0 \times x + 1 \times 1 \end{bmatrix} = \begin{bmatrix}[r] 9 & 16 \ 0 & -y \end{bmatrix} \\[1em] \Rightarrow \begin{bmatrix}[r] 9 + 0 & 3x + x \ 0 + 0 & 0 + 1 \end{bmatrix} = \begin{bmatrix}[r] 9 & 16 \ 0 & -y \end{bmatrix} \\[1em] \Rightarrow \begin{bmatrix}[r] 9 & 4x \ 0 & 1 \end{bmatrix} = \begin{bmatrix}[r] 9 & 16 \ 0 & -y \end{bmatrix} \\[1em]

By definition of equality of matrices we get,

⇒ 4x = 16 and -y = 1
∴  x = 4 and y = -1.

Hence, the values are x = 4 and y = -1.