If A = $\begin{bmatrix$}[r] 2 & x \ 0 & 1 \end{bmatrix} \text{and } B = \begin{bmatrix}[r] 4 & 36 \ 0 & 1 \end{bmatrix}; find the value of x, given that : A² = B.

Given,

⇒ A² = B

$\Rightarrow \begin{bmatrix$}[r] 2 & x \ 0 & 1 \end{bmatrix}\begin{bmatrix}[r] 2 & x \ 0 & 1 \end{bmatrix} = \begin{bmatrix}[r] 4 & 36 \ 0 & 1 \end{bmatrix} \\[1em] \Rightarrow \begin{bmatrix}[r] 2 \times 2 + x \times 0 & 2 \times x + x \times 1 \ 0 \times 2 + 1 \times 0 & 0 \times x + 1 \times 1 \end{bmatrix} = \begin{bmatrix}[r] 4 & 36 \ 0 & 1 \end{bmatrix} \\[1em] \Rightarrow \begin{bmatrix}[r] 4 + 0 & 2x + x \ 0 & 0 + 1 \end{bmatrix} = \begin{bmatrix}[r] 4 & 36 \ 0 & 1 \end{bmatrix} \\[1em] \Rightarrow \begin{bmatrix}[r] 4 & 3x \ 0 & 1 \end{bmatrix} = \begin{bmatrix}[r] 4 & 36 \ 0 & 1 \end{bmatrix}

By definition of equality of matrices we get,

3x = 36
⇒ x = 12.

Hence, x = 12.