/sci/ - Science & Math

It seems that for invertible 2x2 matrices [math]A_i[/math] and [math]X[/math]

[math]\det\big(\sum_{i=1}^n A_i+B\big) - \det\big(\sum_{i=1}^n A_i\big) = \det(B) + r[/math]
where
[math]r = {\mathrm{tr}}\big(\sum_{i=1}^n {\mathrm{adj}}(A_i)\cdot B\big)[/math]

with 'adj' the adjunct matrix (flipping components and dividing by the determinant)

I found the case for 2 matrices on SE and got the others by iterating from 2 to 3 and then guessing.
(Here the 2 matrices case https://math.stackexchange.com/questions/673934/expressing-the-determinant-of-a-sum-of-two-matrices))

Anybody seen this before?
Looks cumbersome to prove by induction but it's a helpful formula.