If A = $\begin{bmatrix$}[r] -1 & 1 \ a & b \end{bmatrix} and A² = I, find a and b.

Given, A² = I

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

By definition of equality of matrices we get,

1 + a = 1
⇒ a = 0.

-1 + b = 0
⇒ b = 1.

Hence, a = 0 and b = 1