# prove that matrix is invertible

• Dec 4th 2012, 11:54 PM
Stormey
prove that matrix is invertible
Hi!

i need to prove that if matrix $A^n$ is invertible, then A is also invertible.

TIA!
• Dec 5th 2012, 01:00 AM
chiro
Re: prove that matrix is invertible
Hey Stormey.

Recall that det(AB) = det(A)*det(B) for operators A and B.
• Dec 5th 2012, 01:04 AM
Deveno
Re: prove that matrix is invertible
it's easier to use chiro's hint if you prove the contrapositive:

if A is singular, so is An
• Dec 5th 2012, 03:04 AM
Stormey
Re: prove that matrix is invertible
we didn't learn determinants yet, so i'm guessing it can be solved without using it.

nevertheless, from what i read in wiki, if matrix A is invetrtible, then its determinant is not 0.
and since i want to prove the contrapositive, i need to assume it's singular, and the its det is equal 0.
so:

$det(A)=0$

i can multiply it by det(A) $n-1$ times, and then say that $det(A^n)=0$
but what conclusion can i draw from it?
does that prove it?

BTW, i have no idea what determinant is, so bare with me... (Giggle)
• Dec 5th 2012, 07:10 AM
Deveno
Re: prove that matrix is invertible
you don't need to use determinants, really.

suppose that A is singular. this means there is some non-zero vector v, with Av = 0.

hence Anv = An-1(Av) = An-1(0) = 0.

so An is singular, as well.
• Dec 5th 2012, 07:41 AM
HallsofIvy
Re: prove that matrix is invertible
To show that $A^n$ is invertible, you must show that there exist matrix B such that $A^nB= BA^n= I$ where I is the identity matrix.
Since A is invertible, there exist a matrix C such that AC= CA= I.
Now, what can you say about $A^nC^n= A^{n-1}(AC)C^{n-1}$ and $C^nA^n= C^{n-1}(CA)(A^{n-1})$?
• Dec 5th 2012, 09:27 AM
Stormey
Re: prove that matrix is invertible
thanks, Deveno.

HallsoiIvy, i need to prove that if p then q, not if q then p.
i managed to prove the other direction, but thanks anyway.