Matrix little demonstration

Hello,

I'm a bit bloqued on a little demontration.

A^4=0 , (I-A)^-1 = I²+A+A²+A³

Here is what i've done:

I=(I-A).(I²+AI+A²I+A³I)

=> I=I³+AI²+A²I²+A³I³-AI²-A²I-A³I-(A^4)I

=> I=I+A²+A³-A²-A³-A^4

=> I=I-A^4

=> A^4=0

But it is prove in reverse...

The right way:

If you know that matrix A^4=0, how to prove that (I-A)^-1 = I²+A+A²+A³

But in ths case (harder I supose), I don't know how to do.

Can you help me?

Re: Matrix little demonstration

Consider

(what are the assumptions that you have forgotten to mention? And for good measure also consider )

.

Re: Matrix little demonstration

Sorry but I have no particular assumptions.

The exercice is presented as such

A^4=0 , (I-A)^-1 = I²+A+A²+A³

It's already a problem for me at the base.

Or what do you mean?

If y consider (I-A).(I+A+A²+A³), I start in the same oposite direction that i've done no?

I still have a misanderstanding with this.

Re: Matrix little demonstration

Re: Matrix little demonstration

Okay, I got t now!

Thanks a lot!

Re: Matrix little demonstration

Hello, Adi0101!

Just run the step in reverse . . .

We have: .

. . . . . . . . .

Add

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

Multiply by

. .

. . .

Re: Matrix little demonstration

Great!

It seems so simple now...

Thanks a lot for this detailed demontration!