Let A = I 2 − 2 M M T , where M is real matrix of order 2 × 1 such that the relation M T M = I 1 holds. If λ is a real number such that the relation A X = λ X holds for some non-zero real matrix X of order 2 × 1 , then the sum of squares of all possible values of λ is equal to:

Step-by-step solution

Given: A = I 2 − 2 M M T ⇒ A 2 = I 2 − 2 M M T I 2 − 2 M M T ⇒ A 2 = I 2 − 2 M M T − 2 M M T + 4 M M T M M T ⇒ A 2 = I 2 − 4 M M T + 4 M M T as M T M = I 1 ⇒ A 2 = I 2 Also, A X = λ X ⇒ A 2 X = λ A X ⇒ X = λ λ X as A 2 = I 2 & A X = λ X ⇒ X = λ 2 X ⇒ X λ 2 − 1 = 0 ⇒ λ 2 = 1 ⇒ λ = ± 1 So, sum of square of all possible values = 2