How to verify if a linear system of symmetrical matrix blocks has solution?
I have the matrix:


* $\left[M\right]\_{p \times p}$, symmetrical
* $\left[G\right]\_{p \times q}$


and then, I would like to solve the following linear system:


$$
\underbrace{\begin{bmatrix}
\left[M\right] & \left[G\right] \\
\left[G^T\right] & \left[0\right]
\end{bmatrix}}\_{\left[A\right]}
\cdot
\underbrace{
\begin{bmatrix}
\left[\mu\right] \\
\left[\lambda \right]
\end{bmatrix}}\_{\left[X\right]}
=
\underbrace{
\begin{bmatrix}
\left[F\_{\mu}\right] \\
\left[F\_{\lambda}\right]
\end{bmatrix}}\_{\left[B\right]}
$$


So far, I found this article [Solve linear system with bordered positive definite matrix](https://mathoverflow.net/questions/365524/solve-linear-system-with-bordered-positive-definite-matrix) that explains how to solve this problem, and in some steps we have to calculate $M^{-1}$ and $H^{-1}$, where


$$H = G^T \cdot M^{-1} \cdot G$$


So, the restrictions to exist a solution are:


* $M^{-1}$ exists $\Leftrightarrow \det{M} \ne 0$
* $H^{-1}$ exists $\Leftrightarrow \det H \ne 0$


So, I would like to know if there's a easier way to known if the system $AX = B$ has a solution than:


* Calculate $\det M$, if it's $= 0$, stop because the system doesn't have solution
* Calculate $M^{-1}$
* Calculate $H = G^T M^{-1} G$
* Calculate $\det H$, and if it's $=0$, the system doesn't have solution.


For my specific problem, a computational problem, I know that besides $M$ being symmetric, it's also positive-definite.
So, I formulated the two hypothesis to get a easier conclusion, but I don't know how to prove it and even if it's true:
Suppose that $M$ is inversible, symmetric and positive-definite, if $\det M \ne 0$, so:


* $\det H = 0 \Leftrightarrow \det G^T \cdot G = 0 $
* $\det H = 0 \Leftrightarrow \det G \cdot G^T = 0 $


And the last question: does always $\det \left(G^T \cdot G\right) = \det \left(G \cdot G^T\right)$?



Conditions to solve linear system with matrix blocks
You find various conditions in Section 3 of the [classical review paper](http://page.math.tu-berlin.de/%7Eliesen/Publicat/BenGolLie05.pdf) by Benzi-Golub-Liesen on this kind of problems, which are known as *saddle-point problems*.


