If A = $\begin{bmatrix$}[r] a & 0 \ 0 & 2 \end{bmatrix}, B = \begin{bmatrix}[r] 0 & -b \ 1 & 0 \end{bmatrix}, M = \begin{bmatrix}[r] 1 & -1 \ 1 & 1 \end{bmatrix} and BA = M², find the values of a and b.

Given,

$\Rightarrow BA = M^2 \\[1em] \Rightarrow \begin{bmatrix$}[r] 0 & -b \ 1 & 0 \end{bmatrix}\begin{bmatrix}[r] a & 0 \ 0 & 2 \end{bmatrix} = \begin{bmatrix}[r] 1 & -1 \ 1 & 1 \end{bmatrix}\begin{bmatrix}[r] 1 & -1 \ 1 & 1 \end{bmatrix} \\[1em] \Rightarrow \begin{bmatrix}[r] 0 \times a + (-b) \times 0 & 0 \times 0 + (-b) \times 2 \ 1 \times a + 0 \times 0 & 1 \times 0 + 0 \times 2 \end{bmatrix} = \begin{bmatrix}[r] 1 \times 1 + (-1) \times 1 & 1 \times (-1) + (-1) \times 1 \ 1 \times 1 + 1 \times 1 & 1 \times (-1) + 1 \times 1 \end{bmatrix} \\[1em] \Rightarrow \begin{bmatrix}[r] 0 & 0 - 2b \ a + 0 & 0 \end{bmatrix} = \begin{bmatrix}[r] 1 - 1 & -1 - 1 \ 1 + 1 & -1 + 1 \end{bmatrix} \\[1em] \Rightarrow \begin{bmatrix}[r] 0 & -2b \ a & 0 \end{bmatrix} = \begin{bmatrix}[r] 0 & -2 \ 2 & 0 \end{bmatrix}

By definition of equality of matrices we get,

-2b = -2
⇒ b = 1.

a = 2.

Hence, a = 2 and b = 1.