# I don't understand this invertible problem?

• Dec 4th 2011, 09:08 PM
elpermic
I don't understand this invertible problem?
Hello all, this was on my practice final exam for my linear algebra class.

Let A and B be two nxn square matricies such that AB is invertible. Prove that homogeneous equation (A^2)x=0 Hint:show that A must be invertible,

The only thing I know is that since A is invertible, by the Inverible MAtrix Theorem, Ax=0 has only trivial solution.

Help!!
• Dec 4th 2011, 09:25 PM
Drexel28
Re: I don't understand this invertible problem?
Quote:

Originally Posted by elpermic
Hello all, this was on my practice final exam for my linear algebra class.

Let A and B be two nxn square matricies such that AB is invertible. Prove that homogeneous equation (A^2)x=0 Hint:show that A must be invertible,

The only thing I know is that since A is invertible, by the Inverible MAtrix Theorem, Ax=0 has only trivial solution.

Help!!

If \$\displaystyle A\$ is invertible, then \$\displaystyle A^2\$ is invertible and so \$\displaystyle A^2x=0\$ has a unique solution, in fact, \$\displaystyle x=0\$. Thus, all you have to show is that \$\displaystyle A\$ is invertible. But, note that since \$\displaystyle AB\$ is invertible \$\displaystyle \det(AB)\ne0\$. But what do you know about the determinant of a product?
• Dec 4th 2011, 09:31 PM
elpermic
Re: I don't understand this invertible problem?
This question was before we had learned about determinants. I don't think the professor will allow us to use the determinant to prove it.

That being said the determinant of a product, for example:

det(AB) will be equal to det(A) x det(B)
• Dec 4th 2011, 10:03 PM
redsoxfan325
Re: I don't understand this invertible problem?
Quote:

Originally Posted by elpermic
This question was before we had learned about determinants. I don't think the professor will allow us to use the determinant to prove it.

That being said the determinant of a product, for example:

det(AB) will be equal to det(A) x det(B)

So if \$\displaystyle \det(AB)\neq0\$, what can you say about \$\displaystyle \det A\$ and \$\displaystyle \det B\$?
• Dec 4th 2011, 10:29 PM
elpermic
Re: I don't understand this invertible problem?
since detA and detB aren't 0, then A is invertible. And A^k is invertible for any k
• Dec 4th 2011, 10:31 PM
redsoxfan325
Re: I don't understand this invertible problem?
Bingo
• Dec 4th 2011, 10:35 PM
elpermic
Re: I don't understand this invertible problem?
Alright thank you very much!

I just hope that I can do well on my final :/
• Dec 4th 2011, 10:36 PM
redsoxfan325
Re: I don't understand this invertible problem?
Good luck!