Let A be a 2 × 2 symmetric matrix such that A [ 1 1 ] = [ 3 7 ] and the determinant of A be 1 . If A − 1 = α A + β I , where I is an identity matrix of order 2 × 2 , then α + β equals _______

Question

Let A be a 2 × 2 symmetric matrix such that A [ 1 1 ​ ] = [ 3 7 ​ ] and the determinant of A be 1 . If A − 1 = α A + β I , where I is an identity matrix of order 2 × 2 , then α + β equals _______

TestHub · Accepted Answer

​ Let A = [ a b ​ b d ​ ] [ a b ​ b d ​ ] [ 1 1 ​ ] = [ 3 7 ​ ] , a d − b 2 = 1 a + b = 3 , b + d = 7 , ( 3 − b ) ( 7 − b ) − b 2 = 1 21 − 10 b = 1 → b = 2 , a = 1 , d = 5 A = [ 1 2 ​ 2 5 ​ ] , A − 1 = [ 5 − 2 ​ − 2 1 ​ ] A − 1 = α A + β I [ 5 − 2 ​ − 2 1 ​ ] = [ α + β 2 α ​ 2 α 5 α + β ​ ] α = − 1 , β = 6 → α + β = 5 ​

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