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## Homework Statement

Let A and B be NxN matrices, and assume that their product C = AB is invertible. Without using determinants, prove that A and B must both be invertible.

## Homework Equations

If a NXN matrix A is invertible:

A

**x**=

**0**has only the trivial solution

**0**.

## The Attempt at a Solution

I believe I have this halfway figured out:

Let's take B to be singular, then there exists an

**x**that is not

**0**such that B

**x**=

**0**. Thus:

C = AB => C

**x**= (AB)

**x**= A(B

**x**) =

**0**.

C would have to be singular, as there exists an

**x**not equal to 0 that makes C

**x**=

**0**. This violates the assumption that C is invertible, thus B must be invertible.

Now, I am not sure where to go with A. I've tried applying the same logic, however:

C = AB => C

**x**= (AB)

**x**= (A

**x**)B =

**0**

does not seem to work. Is there some easy way to prove that if B is invertible then A must be also? If not, how can I alter this logic to prove that A is invertible? Can I somehow use the fact that the product of two invertible matrices must be invertible...and if A is not invertible then C cannot be?