What happens when we prepend a row and column to a regular matrix?

Question

I am working with a matrix $M \in \Bbb R^{n \times n}$ that has non-zero determinant. I then prepend a row and a column to $M$, where each entry is positive, so that $M$ is now $(n+1) \times (n+1)$ and looks like $$\begin{pmatrix} a_{11} & b^T\\ c & M \end{pmatrix}$$ where $a_{11} > 0$ and $b$ and $c$ are vectors consisting of all positive numbers. Does this matrix also have a non-zero determinant?

Answer 1

No, you can very easily generate a counterexample.

For instance, start with the $2 \times 2$ matrix $$M = \begin{bmatrix}2 & 1 \\ 1 & 2\end{bmatrix}$$ which has $\det M = 3 \neq 0$. Then, your $3 \times 3$ matrix is $$M' = \begin{bmatrix}? & ? & ? \\ ? & 2 & 1 \\ ? & 1 & 2\end{bmatrix}.$$

Can you fill in the $?$ entries with positive real numbers to make the first and second columns the same? If so, then $\det M' = 0$.