Answer
Applying theorems 8 and 9, we find that
$x\mapsto Ax$ is not invertible
Work Step by Step
Apply Th.8.
We are given that $A$ is a square $n\times n$ matrix, and
$\mathrm{f}.\quad$ The linear transformation $x\mapsto Ax$ is one-to-one.
is not true.
Then,
$\mathrm{a}. \quad A$ is an invertible matrix.
$\mathrm{i}.\quad$ The linear transformation $x\mapsto Ax$ maps $\mathbb{R}^{n}$ onto $\mathbb{R}^{n}$.
are also false.
So, A is not invertible and $x\mapsto Ax$ does not map $\mathbb{R}^{n}$ onto $\mathbb{R}^{n}$
Since A is not invertible, by Th.9, the transformation
$x\mapsto Ax$
is not invertible.
