Let A = 1 10 3 10 - 3 10 1 10 and B = 1 - i 0 1 , where i = - 1 . If M = A T BA , then the inverse of the matrix AM 2023 A T is

Step-by-step solution

Given, A = 1 10 3 10 - 3 10 1 10 ,   B = 1 - i 0 1 AA T = 1 10 3 10 - 3 10 1 10 1 10 - 3 10 3 10 1 10 = 1 0 0 1 B 2 = 1 - i 0 1 1 - i 0 1 = 1 - 2 i 0 1 B 3 = 1 - 2 i 0 1 1 - i 0 1 = 1 - 3 i 0 1 So, we can write B 2023 = 1 - 2023 i 0 1 M = A T BA (given) M = AA T B = 1 0 0 1 1 - i 0 1 = 1 - i 0 1 M 2 = M · M = A T BAA T BA = A A T . AA T . B 2 = AA T B 2 M 3 = AA T B 3 M 2023 = AA T B 2023 = B 2023 ( ∵   A A T is an identity matrix) ∴   M 2023 = B 2023 = 1 - 2023 i 0 1 Inverse of AM 2