Let A be a 2 × 2 matrix with real entries. Let I be the 2 × 2 identity matrix. Denote by tr ( A ) , the sum of diagonal entries of A . Assume that A 2 = 1 . Statement -1: If A  = 1 and A  = − 1 , th

Question

Let A be a 2 × 2 matrix with real entries. Let I be the 2 × 2 identity matrix. Denote by tr ( A ) , the sum of diagonal entries of A . Assume that A 2 = 1 . Statement -1: If A  = 1 and A  = − 1 , then det A = − 1 . Statement − 2 : If A  = 1 and A  = − 1 , then tr ( A )  = 0 .

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Let A = [ a c ​ b d ​ ] so that A 2 = [ a 2 + b c a c + d c ​ ab + b d b c + d 2 ​ ] = [ 1 0 ​ 0 1 ​ ] ⇒ a 2 + b c = 1 = b c + d 2 and ( a + d ) c = 0 = ( a + d ) b . Since A  = I , A  = 1 , a = − d and hence det A = ​ 1 − b c ​ c ​ b − 1 − b c ​ ​ ​ = − 1 + b c − b c = − 1 Statement 1 is true. But tr. A = 0 and hence statement 2 is false.

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