The number of 2 × 2 matrices A with real entries, such that A + A T = 3 I and A A T = 5 I , is equal to

Question

The number of 2 × 2 matrices A with real entries, such that A + A T = 3 I and A A T = 5 I , is equal to, 0, 1, 2, Infinite

MARKS App · Accepted Answer

Let, A = a b c d , A T = a c b d Given, A + A T = 3 I 2 a b + c b + c 2 d = 3 0 0 3 ⇒ a = 3 2 , d = 3 2 , b = - c A = 3 2 b - b 3 2 A A T = 3 2 b - b 3 2 3 2 - b b 3 2 ⇒ 9 4 + b 2 0 0 9 4 + b 2 = 5 0 0 5 ⇒ 9 4 + b 2 = 5 ⇒ b 2 = 5 - 9 4 ⇒ b = ± 11 2 Hence, 2 matrices are possible

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