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

JEE Main · 2022 · 26 Jul 2022 · Shift 2 · Matrices

The number of matrices

A = [a c b d]

, where

a, b, c, d \in {- 1, 0, 1, 2, 3, \dots, 10}

, such that

A = A^{- 1}

, is ______.

Answer: 50

A = A^{- 1}

, then

A^{2} = I

. Computing:

[a^{2} + b c a c + c d ab + b d b c + d^{2}] = [1001]

This gives:

a^{2} + b c = 1

ab + b d = 0

a c + c d = 0

b c + d^{2} = 1

.

From the first and fourth equations:

a^{2} = d^{2}

, so

a = d

a = - d

.

**Case I:

a + d = 0

$(a, d) = (- 1, 1)$ : $b c = 0$ . Either $b = 0$ (12 choices for $c$ ) or $c = 0$ (12 choices for $b$ ), minus the overlap $(0, 0)$ : 23 pairs.
$(a, d) = (1, - 1)$ : Similarly, 23 pairs.
$(a, d) = (0, 0)$ : $b c = 1$ , so $(b, c) \in {(1, 1), (- 1, - 1)}$ : 2 pairs.

**Case II:

a = d

From equations (2) and (3), if $a \neq = 0$ , then $b = c = 0$ and $a^{2} = 1$ .
$(a, d) \in {(1, 1), (- 1, - 1)}$ : 2 pairs.

Total:

23 + 23 + 2 + 2 = 50

JEE Main · 2022 · 26 Jul 2022 · Shift 2 · Integer answer