Why Linear Differential Equations are First Order

First, a note on format. Dⁿ(y)_x is the n-th differential of y with respect to x. 
I've used this format to keep it somewhat close to the format used by sagemath. However, I have broken down here and there and used y` to represent the first differential of y w.r.t. x.

It's a well-known fact that linear differential equations of the form:

Dⁿ(y)_x+ ... + a D(y)_x + by =f(x)

are particularly easy to solve.

Since:

Dⁿ(y+z)_x = Dⁿ(y)_x + Dⁿ(z)_x 

it is easy to split the problem into several subproblems and add them all at the end.

But matrices allow us to rearrange linear differential equations of any order into a simple first order differential equation of the form:

D(y)_x = ky

Matrix Trace, Determinants and Eigenvalues

This is the start of a quick series of posts showing some powerful properties (and proofs) of matrix eigenvalues. It is assumed that the reader can calculate eigenvectors/values.

However, the series refers to some basic algebra and matrix properties that, while used everywhere, usually don't have accessible or comprehensive  proofs available on the Internet (You can find them in good linear algebra textbooks). It is also a good place to show some neat tricks for sagemath.
There are better and more comprehensive proofs out there, but these are a good start.

We will start today with matrix traces and determinants, and their relationship to matrix eigenvalues. Basically we will be proving that for any n x n matrix, the eigenvalues add up to the sum of the diagonal elements, and multiply up to the determinant of the matrix.