Let A be an n times n matrix with eigenvalue 1 In be the n t

Let A be an n times n matrix with eigenvalue -1. I_n be the n times n identity matrix and 0_n be the n times n zero matrix. Which of the following are true? 1 (- l)^k is an eigenvalue of A^k for all k Element N. I_n + A is singular. I_n + A = 0_n. If x Element R^n such that Ax = -x, then x = 0.

Solution

(1) is correct   

when the eigen value \' k \' for the matrix exist then

Ax= kx here given k=-1

Ax= - 1 x

A²x = A(Ax) = A ( -1X ) = - A x= - ( 1x) = (- 1)² x

ie for the matrix A² te eigen value is ( - 1)²

when we proceed like this \' k \' times we get   

for the matrix A^k  the eigen value is (- 1)^k