QJEEM22Ma7179: JEE Main 2022 Mathematics - Class 12 JEEM22Ma7179 Solution: The number of matrices , where , such that , is ___________.

$\because$ $$A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$$ then $$A^2=\left[\begin{array}{cc}{a}^2+bc& b(a+d)\\ c(a+d)& bc+d^2\end{array}\right]$$

For A ^{$-$ 1} must exist $ad - bc \ne 0$ ...... (i)

and $A = {A^{ - 1}} \Rightarrow {A^2} = I$

$\therefore$ ${a^2} + bc = {d^2} + bc = 1$ ...... (ii)

and $b(a + d) = c(a + d) = 0$ ...... (iii)

Case I : When a = d = 0, then possible values of (b, c) are (1, 1), ( $-$ 1, 1) and (1, $-$ 1) and ( $-$ 1, 1).

Total four matrices are possible.

Case II : When a = $-$ d then (a, d) be (1, $-$ 1) or ( $-$ 1, 1).

Then total possible values of (b, c) are $(12 + 11) \times 2 = 46$ .

$\therefore$ Total possible matrices $= 46 + 4 = 50$ .