Exercise 11

Suppose that $A$ and $B$ are two $n\times n$ matrices and that $AB$ is invertible. We would like to explain why both $A$ and $B$ are invertible.

1. We first explain why $B$ is invertible.

   1. Since $AB$ is invertible, explain why any solution to the homogeneous equation $AB\xvec = \zerovec$ is $\xvec=\zerovec$.

   2. Use this fact to explain why any solution to $B\xvec = \zerovec$ must be $\xvec=\zerovec$.

   3. Explain why $B$ must be invertible.

2. Now we explain why $A$ is invertible.

   1. Since $AB$ is invertible, explain why the equation $AB\xvec=\bvec$ is consistent for every vector $\bvec$.

   2. Using the fact that $AB\xvec = A(B\xvec) = \bvec$ is consistent for every $\bvec$, explain why every equation $A\xvec = \bvec$ is consistent.

   3. Explain why $A$ must be invertible.