Can a non-invertible matrix be extended to an invertible one? For a square matrix $M$ call any square matrix M' of the form 
$$\left(\begin{array}{cc}
M & A\\
B & C
\end{array}\right)$$
an extension of $M$. Does it follow that if $M$ is not invertible that all extensions $M'$ are not invertible? I believe the answer is no. If not, is there an extension that is invertible? Can we prove that there always is?
 A: For any $M$, the matrix
$$\pmatrix{M&I\\I&0}$$
is invertible.
A: My answer is similar in spirit to Lord Shark the Unknown's, but provides a few more details.
First of all, as Lord Shark affirms, for any matrix square matrix $M$, the extension matrix $E_M$,
$E_M = \begin{bmatrix} M & I \\ I & 0 \end{bmatrix}, \tag 1$
is invertible.  The easiest way to see this is to show that 
$\ker E_M = \{ 0 \}; \tag 2$
now if
$\text{size}(M) = n, \tag 3$
that is, $M$ is an $n \times n$ matrix over some field $\Bbb F$, then
$\text{size}(E_M) = 2n; \tag 4$
thus $E_M$ may be considered as operating on the $2n$-dimensional vector space $\Bbb F^{2n}$, any vector $v \in \Bbb F^{2n}$ of which may be written in "stacked form"
$v = \begin{pmatrix} x \\ y \end{pmatrix}, \tag 5$
where $x, y \in \Bbb F^n$; then if 
$E_M v = 0, \tag 6$
we have
$\begin{pmatrix} Mx + y \\ x \end{pmatrix} = \begin{bmatrix} M & I \\ I & 0 \end{bmatrix}  \begin{pmatrix} x \\ y \end{pmatrix} = 0, \tag 7$
from which we conclude
$x = 0, \tag 8$
and 
$y = -Mx = -M(0) = 0; \tag 9$
thus
$v = 0, \tag{10}$
which shows that (2) binds and thus that $E_M$ is invertible.  We may in fact find the inverse $E_M^{-1}$ by setting 
$E_M v = w, \tag{11}$
where
$w = \begin{pmatrix} s \\ t \end{pmatrix}; \tag{12}$
then as above we have
$Mx + y = s, \; x = t, \tag{13}$
whence
$y = s - Mt; \tag{14}$
(13) and (14) together show that
$E_M^{-1} = \begin{bmatrix} 0 & I \\ I & -M \end{bmatrix}, \tag{15}$
which we may easily check:
$E_M^{-1} w = \begin{bmatrix} 0 & I \\ I & -M \end{bmatrix} \begin{pmatrix} s \\ t \end{pmatrix} = \begin{pmatrix} t \\ s - Mt \end{pmatrix} = \begin{pmatrix} x \\ y \end{pmatrix}, \tag{16}$
using (13) and (14).
