JEE Main · 2021 · 26 Feb 2021 · Shift 1 · Matrices

Let A be a symmetric matrix of order 2 with integer entries. If the sum of the diagonal elements of

A^{2}

is 1 , then the possible number of such matrices is :

Answer: (C)

Solution

Write

A = (p q q s)

with

p, q, s

integers. Then

A^{2} = (p^{2} + q^{2} q (p + s) q (p + s) q^{2} + s^{2})

so the sum of the diagonal entries of

A^{2}

p^{2} + 2 q^{2} + s^{2} = 1

All three squares are non-negative integers, and

2 q^{2} \leq 1

forces

q = 0

, leaving

p^{2} + s^{2} = 1

, whose integer solutions are

(p, s) = (1, 0), (- 1, 0), (0, 1), (0, - 1)

.
The number of such matrices is

4

JEE Main · 2021 · 26 Feb 2021 · Shift 1 · Single correct option