# Thread: I don't understand this invertible problem?

1. ## 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!!

2. ## Re: I don't understand this invertible problem?

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?

3. ## 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)

4. ## Re: I don't understand this invertible problem?

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 $\displaystyle \det B$?

5. ## 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

Bingo

7. ## Re: I don't understand this invertible problem?

Alright thank you very much!

I just hope that I can do well on my final :/

Good luck!