Say $A$ is a $n$ by $n$ positive definite matrix. Let $b$ be a column vector in $\mathbb{R}^n$.
Consider the following quantity: $$b^TA^*b$$ where $A^*$ is the cofactor matrix of $A$. A simple calculation reveals that this quantity is (up to a sign) the same as $$det\left(\begin{align}A ,& b \\b^T, &0 \end{align}\right).$$
Now is there a relationship between this quantity and the determinant of $A$? By relationship, I mean one controls the other somehow. I am asking this because I was told that there might be a formula for determinants in terms of the above quantity. But the person who told me this is not sure and he doesn't remember where he saw it.
Any sort of reference towards this direction is highly appreciated! Thanks.
Consider $\det \begin{bmatrix} A & b \\ b^T & 0 \end{bmatrix} = \det \begin{bmatrix} I & 0 \\ -b^T A^{-1} & 1 \end{bmatrix} \det \begin{bmatrix} A & b \\ b^T & 0 \end{bmatrix} = \det \left( \begin{bmatrix} I & 0 \\ -b^T A^{-1} & 1 \end{bmatrix} \begin{bmatrix} A & b \\ b^T & 0 \end{bmatrix} \right)$ yields
$\det \begin{bmatrix} A & b \\ b^T & 0 \end{bmatrix} = -b^T A^{-1} b \det A $.