If $\begin{bmatrix$}[r] -1 & 0 \ 0 & 1 \end{bmatrix} \begin{bmatrix}[r] a & b \ c & d \end{bmatrix} = \begin{bmatrix}[r] 1 & 0 \ 0 & -1 \end{bmatrix}, find a, b, c and d.

Given,

$\begin{bmatrix$}[r] -1 & 0 \ 0 & 1 \end{bmatrix} \begin{bmatrix}[r] a & b \ c & d \end{bmatrix} = \begin{bmatrix}[r] 1 & 0 \ 0 & -1 \end{bmatrix} \\[0.5em] \Rightarrow \begin{bmatrix}[r] -1 \times a + 0 \times c & -1 \times b + 0 \times d \ 0 \times a + 1 \times c & 0 \times b + 1 \times d \end{bmatrix} = \begin{bmatrix}[r] 1 & 0 \ 0 & -1 \end{bmatrix} \\[0.5em] \Rightarrow \begin{bmatrix}[r] -a & -b \ c & d \end{bmatrix} = \begin{bmatrix}[r] 1 & 0 \ 0 & -1 \end{bmatrix} \\[0.5em]

By definition of equality of matrices we get,

⇒ -a = 1 or a = -1

⇒ -b = 0 or b = 0

⇒ c = 0

⇒ d = -1.

Hence, the value of a = -1, b = 0, c = 0 and d = -1.