If $\begin{bmatrix$}[r] 1 & 2 \ 3 & 3 \end{bmatrix} \begin{bmatrix}[r] x & 0 \ 0 & y \end{bmatrix} = \begin{bmatrix}[r] x & 0 \ 9 & 0 \end{bmatrix}, find the values of x and y.

Given,

$\begin{bmatrix$}[r] 1 & 2 \ 3 & 3 \end{bmatrix} \begin{bmatrix}[r] x & 0 \ 0 & y \end{bmatrix} = \begin{bmatrix}[r] x & 0 \ 9 & 0 \end{bmatrix} \\[1em] \Rightarrow \begin{bmatrix}[r] 1 \times x + 2 \times 0 & 1 \times 0 + 2 \times y \ 3 \times x + 3 \times 0 & 3 \times 0 + 3 \times y \end{bmatrix} = \begin{bmatrix}[r] x & 0 \ 9 & 0 \end{bmatrix} \\[1em] \Rightarrow \begin{bmatrix}[r] x + 0 & 0 + 2y \ 3x + 0 & 0 + 3y \end{bmatrix} = \begin{bmatrix}[r] x & 0 \ 9 & 0 \end{bmatrix} \\[1em] \Rightarrow \begin{bmatrix}[r] x & 2y \ 3x & 3y \end{bmatrix} = \begin{bmatrix}[r] x & 0 \ 9 & 0 \end{bmatrix} \\[1em]

By definition of equality of matrices,

⇒ 2y = 0, 3x = 9 and 3y = 0
⇒ y = 0, x = 3 and y = 0.

Hence, the values are x = 3 and y = 0.