# Inverse Matrix

#### Yankel

##### Active member
Hello,

I have this question, I need to choose the correct answer:

A is a square matrix such that

$A^{2}+A+I=0$

a) $A^{-1}=A$

b) $A^{-1}=A^{2}$

c) It is not possible to say if A is invertible

d) A is not invertible

e) $A^{-1}=A+I$

I got that

$-A^{2}-A=I$

and thus

$A(-A-I)=0$

and thus

$A^{-1}=(-A-I)$

and answer which doesn't exist, am I wrong ?

Thanks !

#### HallsofIvy

##### Well-known member
MHB Math Helper
Yes, you are wrong! You seem to be under the impression that if B is the inverse of A then AB= 0. That is not the case- it is AB= I.
From A^2+ A+ I= 0, -(A^2+ A)= I, A(-(A- I))= I.

#### Yankel

##### Active member
Putting 0 was a typing mistake. You seemed to be doing what I did, your answer also doesnt appear as an option

#### Ackbach

##### Indicium Physicus
Staff member
Perhaps the original equation was $A^{2}+A-I=0$? Your working (except for the typo) seems correct to me.

#### Yankel

##### Active member
no, the question is as I wrote it, so you guys agree with me that none of the possible answers is correct ?

#### Ackbach

##### Indicium Physicus
Staff member
no, the question is as I wrote it, so you guys agree with me that none of the possible answers is correct ?
As stated, I would agree that none of the answers are correct.

#### Prove It

##### Well-known member
MHB Math Helper
I think the answer must be c), there is never any guarantee that an arbitrary square matrix is invertible.

#### Deveno

##### Well-known member
MHB Math Scholar
I think the answer must be c), there is never any guarantee that an arbitrary square matrix is invertible.
Any square matrix that satisfies a polynomial with a non-zero constant term *is* invertible.

This is equivalent to saying the minimal polynomial for such a matrix has non-zero constant term, that is: it does not have 0 as an eigenvalue and thus has trivial kernel.