Answer
Applying Th.8(k. and a.) and the associativity of matrix multiplication,
$\mathrm{A}$ is invertible.
Work Step by Step
Let $\mathrm{C}$ be the inverse of $AB$.
Then, by definition of inverse matrix,
$(AB)\mathrm{C}=I$
Since matrix multiplication is associative
$(AB)\mathrm{C}=A(B\mathrm{C})=I$.
What we have is
$\mathrm{k}.\quad$ There is an $n\times n$ matrix $D=\mathrm{B}\mathrm{C}$ such that $AD=I$.
is true, which means that, by Th.8,
$\mathrm{a}. \quad A$ is an invertible matrix.
is also true.
