If A,B are nxn and AB is invertible, then A and B are invertible

[Mr Davis 97]

Homework Statement

Let A and B be nxn matrices such that AB is invertible. Show that A and B are also invertible.

Homework Equations

The Attempt at a Solution

I feel that there are many ways to do this. My first idea was to use the fact that CAB = I and ABC = I for some C, which implies that (CA)B = I and A(BC) = I, which means that A has a right inverse and B has a left inverse. But since A and B are nxn, BC and CA are not just right and left inverses, respectively, but they are the inverses of A and B. Is this correct reasoning?

Also, would it be any better to make use of the isomorphism from matrices and linear transformations, prove the result for linear transformations, and hence prove the result for matrices?

 

[Ray Vickson]

Homework Statement

Let A and B be nxn matrices such that AB is invertible. Show that A and B are also invertible.

Homework Equations

The Attempt at a Solution

I feel that there are many ways to do this. My first idea was to use the fact that CAB = I and ABC = I for some C, which implies that (CA)B = I and A(BC) = I, which means that A has a right inverse and B has a left inverse. But since A and B are nxn, BC and CA are not just right and left inverses, respectively, but they are the inverses of A and B. Is this correct reasoning?

Also, would it be any better to make use of the isomorphism from matrices and linear transformations, prove the result for linear transformations, and hence prove the result for matrices?

Do you know about determinants?

 

[Mr Davis 97]

Do you know about determinants?

I know about them, but in this course we are not allowed to use them yet.

 

[fresh_42]

You could show, that the kernels of $A$ and $B$ have to be $\{0\}$ and apply the dimension formula $\dim \ker C + \dim \textrm{im}\, C = n$ to show surjectivity.

 

[pasmith]

Homework Statement

Let A and B be nxn matrices such that AB is invertible. Show that A and B are also invertible.

Homework Equations

The Attempt at a Solution

I feel that there are many ways to do this. My first idea was to use the fact that CAB = I and ABC = I for some C, which implies that (CA)B = I and A(BC) = I, which means that A has a right inverse and B has a left inverse. But since A and B are nxn, BC and CA are not just right and left inverses, respectively, but they are the inverses of A and B. Is this correct reasoning?

This is correct; it is also the easiest way to prove the result.