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Polynomial problem proof?

delgeezee

New member
May 24, 2013
9
Let A be a square matrix,

a) show that \(\displaystyle (I-A)^{-1}= I + A + A^2 + A^3 if A^4 = 0\)

b) show that \(\displaystyle (I-A)^{-1}= I + A + A^2+.....+A^n \) if \(\displaystyle
A^{n+1}= 0\)
 

Amer

Active member
Mar 1, 2012
275
Re: polynomial problem proof?

B is the inverse of A iff [tex] AB = BA = I [/tex]
so
try
[tex] (I -A )( I + A + A^2 + A^3) = I-A + A - A^2 + A^2 - A^3 + A^3 - A^4 = I - A^4 = I [/tex]