Prove Invertibility of nxn Matrix AB=I

[sheldonrocks97]

Homework Statement

Prove that every n x n matrix A for which there exists an n x n matrix B such that AB = I must be invertible. Hint: Use properties of determinants.

Homework Equations

None that I am aware of.

The Attempt at a Solution

I tried finding the inverse of the matrix and multiplying by an elementary matrix. I also tried finding the determinants of a simple matrix and using it's properties but nothing is working :(

 

[pasmith]

Homework Statement

Prove that every n x n matrix A for which there exists an n x n matrix B such that AB = I must be invertible. Hint: Use properties of determinants.

Homework Equations

None that I am aware of.

The Attempt at a Solution

I tried finding the inverse of the matrix and multiplying by an elementary matrix. I also tried finding the determinants of a simple matrix and using it's properties but nothing is working :(

You are supposed to use the fact that $\det (AB) = (\det A) (\det B)$.