MHB Can (I-A)^{-1} Be Expressed as a Series When A^4 = 0?

Thread starter delgeezee
Start date Jun 11, 2013
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Polynomial Proof

Jun 11, 2013

delgeezee

Let A be a square matrix,

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

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

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Jun 11, 2013

Amer

Re: polynomial problem proof?

B is the inverse of A iff AB = BA = I
so
try
(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

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