Thread: 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!!

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 is invertible, then is invertible and so has a unique solution, in fact, . Thus, all you have to show is that is invertible. But, note that since is invertible . But what do you know about the determinant of a product?

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)

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 , what can you say about and ?

since detA and detB aren't 0, then A is invertible. And A^k is invertible for any k

Bingo

Alright thank you very much!

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

Good luck!