Matrix manipulation (inverse, lin. alg.)

Homework Statement

Let A$\in$M$_{n}$($\Re$) a matrix verifying

A$^{3}$-A$^{2}$-I$_{n}$=0

a) Show that A is inversible and calculate it
b) Show that the solution X$\subset$M$_{n}$($\Re$) of the equation

A$^{k}$(A-I$_{n}$)X=I$_{n}$

has a unique solution.

The Attempt at a Solution

I'm having trouble with starting this one. I'm quite rubbish with these matrices in linear algebra, but I have exams in a few days and this question was on it, so i need help!

I know the criteria for matrix inverse (AB=BA=I). However there's too much going on... help me dissect it? thanks a lot to anyone for any help, much appreciated.

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HallsofIvy
Homework Helper

Homework Statement

Let A$\in$M$_{n}$($\Re$) a matrix verifying

A$^{3}$-A$^{2}$-I$_{n}$=0

a) Show that A is inversible and calculate it
This is kind of trivial! $A^3- A^2= A(A^2- A)= (A^2- A)A= I$.

b) Show that the solution X$\subset$M$_{n}$($\Re$) of the equation

A$^{k}$(A-I$_{n}$)X=I$_{n}$

has a unique solution.
Again, from $A^3- A^2- I= 0$, we have $A^2(A- I)= I$ so for any $k\ge 2$, $A^k(I- I)X= A^{k-2}(A^2(A- I)X= A^{k- 2}X= I$. And since A has an inverse, you just multiply both sides by $A^{-1}$ k- 2 times. The cases where k= 0 or k= 1 are simple.

The Attempt at a Solution

I'm having trouble with starting this one. I'm quite rubbish with these matrices in linear algebra, but I have exams in a few days and this question was on it, so i need help!

I know the criteria for matrix inverse (AB=BA=I). However there's too much going on... help me dissect it? thanks a lot to anyone for any help, much appreciated.
There was really no "matrix" algebra involved here, just the definitions and basic algebraic manipulation.

Oh I see, that was easier than I thought. I always get tripped up when they mention matrices and think that I have to tread very cautiously.

thank you

Actually, can you help me out with the last steps for solving? I'm still a bit caught up. I attached my attempt where you left off.

thanks

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