Let B = 1 2 0 1 and A be a 2 × 2 matrix satisfying A T - 1 = A . If X = A B A T , then A T X 2021 A =

Step-by-step solution

Given A T - 1 = A ,   ⇒ A A T = A T A = I . Now, X = A B A T ⇒ X 2021 = A B A T 2021 ⇒ X 2021 = A B A T A B A T A B A T . . . A B A T We know that, the matrix multiplication is associative i.e. for three matrices, we have P Q R = P Q R . ⇒ X 2021 = A B A T A B A T A B A T A . . . A T A B A T Using the given condition, we get ⇒ X 2021 = A B I B I B I . . . I B A T ⇒ X 2021 = A B B B . . . B A T ⇒ X 2021 = A B 2021 A T ⇒ A T X 2021 A = A T A B 2021 A T A ⇒ A