A First Course in Linear Algebra · Problem MINM.T25

Matrices — Robert Beezer

Mathematics — Linear Algebra · matrix-powers-invertibility

Problem

The notation

A^k

means a repeated matrix product between

k

copies of the square matrix

A

. 1. Assume

A

is an

n \times n

matrix where

A^2 = O

(which does not imply that

A = O

.) Prove that

I_n - A

is invertible by showing that

I_n + A

is an inverse of

I_n - A

. 2. Assume that

A

is an

n \times n

matrix where

A^3 = O

. Prove that

I_n - A

is invertible. 3. Form a general theorem based on your observations from parts (1) and (2) and provide a proof.