About a determinant identity.

Question

If $A$ is any matrix and $B$ is a rank $2$ matrix of the same dimension then it follows that for any real $t$,

$det(A -B) = [1-\partial_p + \frac{1}{2}\partial_p^2 ]det(A + pB) \vert _{p=0}$

I guess this can be shown by writing $B$ as a sum of two rank $1$ matrices and then using this formula ( http://en.wikipedia.org/wiki/Matrix_determinant_lemma ) and then doing the Sherman-Morrison for the inverse and then again doing the matrix determinant lemma. Eventually this leads to the conclusion that $det(A+tB)$ is a polynomial quadratic in $t$. And then this differential operator follows from writing the Taylor series of $det(A+tB)$ as that of a quadratic polynomial in $t$ and then setting $t=-1$

• Firstly is the argument above correct?

• Secondly are there other differential operators which can do such a similar mapping?

Answer 1

Spoiler hints :

• $A\to \textrm{det}(A)$ is an alternated multilinear map if you see a matrix as the data of $n$ vectors (its columns for instance), that is, if you identify $\mathbf{M}_n(\mathbf{R})$ and $\underbrace{\mathbf{R}^n \times \ldots \times \mathbf{R}^n}_{\textrm{$n$ times}}$
• do you know of the calculate the differential of an alternated multilinear map ?
• do you you know the differential of $g\circ f$ in function of differentials of $g$ and $f$ ?

This will allow you to calculate easily the derivatives of $t\mapsto \textrm{det}(A+tB)$ (which is a polynomial function in the variable $t$) and to conclude.