I am looking for an expression that convert calculation of given quadratic form into determinant of some block matrix. I see this in that form (that may be incorrect):


$x^T A x = \begin{vmatrix} x^T & 0 & 0 \\ 0 & A & 0 \\ 0 & 0 & x \end{vmatrix}$


To clarify. The above matrix is a block matrix with “additional” block entries on the diagonal of the matrix. There are two 3-dimensional vectors $x$ and a matrix $A$ of size $3\times3$.


My first port of call was the application of the determinant formula for block matrices, given in the next link:


<https://en.wikipedia.org/wiki/Determinant#Block_matrices>


Firstly, I did not find the application of the above formulas to the structure with which I am dealing, and secondly, the standard det commands in Mathkad and Matlab do not work with such structures. I was wondering if there was an article among mathematicians or a well-known formula that would facilitate this task.


The task for me now is to somehow find an expression for the determinant, so that the solution to the problem consists of at least a combination of “det” commands Mathkad and Matlab.



Determinant of a block matrix with dissimilar elements
It's hard for me to understand what you are asking, but maybe this answers your question. One has
$$
-x^TAx = M / A^{-1},
$$
where
$$
M = \begin{bmatrix}A^{-1} & x\\ x^T & 0 \end{bmatrix}
$$
and the symbol $/$ denotes the [Schur complement](http://en.wikipedia.org/wiki/Special:Search?search=Schur%20complement).


One has
$$
\det(M / A^{-1}) = \det(M) / \det(A^{-1}) = \det(M)\det(A),
$$
so you get
$$
x^TAx = -\det(M)\det(A).
$$


Side note: if your goal is actually computing things numerically, then using determinants is usually considered a bad idea. The good thing about determinants is that they give you closed forms of stuff, but when it comes to scientific computing you are almost always better off without using them.


