Show that the matrix is invertible.

Question

Suppose $A$ be an $n×n$ real matrix be such that $A^k= O$ for some $k \in \Bbb N$ then prove that $I+A$ is invertible.

Please do not use eigenvalues.

score 3 · Answer 1 · answered Mar 17 '16 at 19:38

3

$I-A^k = (I+A)(I - A...+(-1)^{k-1} A^{k-1}) = I$

$(I - A...+(-1)^{k-1} A^{k-1}) = (I+A)^{-1}$

answered Mar 17 '16 at 19:38

Doug M

score 2 · Answer 2 · answered Mar 17 '16 at 19:19

2

1 - A + A^2 - A^3 + ... (a finite sum)

answered Mar 17 '16 at 19:19

almagest

Typo: first one should be an identity matrix, not a 1. – bartgol Mar 17 '16 at 19:38

2 Answers2