# Characteristic Eq for Matrix problem

• Zoe-b
In summary, the coefficient of x in the expansion of det(x*I-A) can be found by taking the sum from i = 1 to n of the determinant of the Aii minor, where Aii is the submatrix of A formed by deleting row i and column i. This can also be expressed as the sum of the product of eigenvalues divided by each eigenvalue of A. This method can be used for diagonal and diagonalizable matrices.

## Homework Statement

Where A is an n x n matrix and I is the n x n identity
In the expansion of det(x*I-A), show that the coefficient of x is equal to the sum from i = 1 to n of the determinant of the Aii minor. (where Aii = the submatrix of A formed by deleting row i and column i)

## Homework Equations

if X1, X2, ... Xn are the eigenvalues of A then the coefficient of x is equal to the sum from i = 1 to n of (product of eigenvalues)/Xi.

## The Attempt at a Solution

I have tried quite a few different ways of doing this but getting nowhere..
using the definition of a determinant as a sum of products over permuations of Sym (n) and then using the Leibnix rule for integrals I seem to find that (where a11 is the top left entry of A and so on)
d(det(x*I-A)/dx = sum (from i =1 to n) of det(x*I-A)/(x-aii)

The constant coefficient of this will be equal to the x coefficient of det(x*I-A) but the RHS only cancels directly to what I want if aii is an eigenvalue, which obviously isn't always the case. I tried to use row permutations but this obviously changes the Aii minors and so doesn't seem to work. Any ideas? I was trying to follow the proof on

(sorry for the long link!) but the notation is a bit beyond me. Am I on the right track at all or is there a better way to attempt this? Sorry for not including full working but without the maths language its near impossible to be clear.
Thanks

Can you prove it for when A is diagonal

Yes I understand for when it is diagonal but altho row/column operations of form (row i -> row i + d*row j) won't change the determinant of the matrix, they will change the determinant of the minors.. I don't see how to relate the diagonal form back to the original matrix.

How about for a diagonaliable matrix? That is the key here.

Hmmmn I'm still pretty clueless but I'll give it another go tomorrow! Thanks