Question

The number of matrices
$A=\begin{pmatrix} a & b \\\ c & d \\\ \end{pmatrix}$, where a,b,c,d ∈−1,0,1,2,3,…..,10
such that A = A-1, is ______.

Answer 1

The correct answer is 50
$∵ A=\begin{bmatrix} a & b \\\ c & d \\\ \end{bmatrix}$ then $A^2=\begin{bmatrix} a^2+bc & b(a+d) \\\ c(a+d) & bc+d^2 \\\ \end{bmatrix}$
For A –1 must exist ad – bc ≠ 0 …(i)
and A = A –1⇒ A 2 = I
∴ 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) × 2 = 46.
∴ Total possible matrices = 46 + 4 = 50.