If a matrix is invertible, how do we prove that its principal submatrices are invertible?
Solution 1:
The claim you are looking to prove is false, at least by the definition of principal submatrix I am familiar with. Consider as a counterexample:
$$A=\left[\begin{matrix}1&1&1\\1&1&0\\1&0&0\end{matrix}\right]$$
Now $A$ is clearly invertable, but has as a principal submatrix the non-invertable $B$:
$$B=\left[\begin{matrix}1&1\\1&1\end{matrix}\right]$$