$$\because$$ $$A = \left[ {\matrix{ a & b \cr c & d \cr } } \right]$$ then $${A^2} = \left[ {\matrix{ {{a^2} + bc} & {b(a + d)} \cr {c(a + d)} & {bc + {d^2}} \cr } } \right]$$

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 $$= 46 + 4 = 50$$.