The number of matrices $$A=\left(\begin{array}{ll}a & b \\ c & d\end{array}\right)$$, where $a, b, c, d \in\{-1,0,1,2,3, \ldots \ldots, 10\}$, such that $A=A^{-1}$, is ___________.

<p>$\because$ $A = \left[ {\matrix{ a & b \cr c & d \cr } } \right]$ then $${A^2} = \left[ {\matrix{ {{a^2} + bc} & {b(a + d)} \cr {c(a + d)} & {bc + {d^2}} \cr } } \right]$$</p> <p>For A^$-$1 must exist $ad - bc \ne 0$ ...... (i)</p> <p>and $A = {A^{ - 1}} \Rightarrow {A^2} = I$</p> <p>$\therefore$ ${a^2} + bc = {d^2} + bc = 1$ ...... (ii)</p> <p>and $b(a + d) = c(a + d) = 0$ ...... (iii)</p> <p>Case I : When a = d = 0, then possible values of (b, c) are (1, 1), ($-$1, 1) and (1, $-$1) and ($-$1, 1).</p> <p>Total four matrices are possible.</p> <p>Case II : When a = $-$d then (a, d) be (1, $-$1) or ($-$1, 1).</p> <p>Then total possible values of (b, c) are $(12 + 11) \times 2 = 46$.</p> <p>$\therefore$ Total possible matrices $= 46 + 4 = 50$.</p>