Suppose you have a squared matrix with rows and columns. Is there a relationship between its eigenvalues and its determinant? And what about its trace?

#### Determinant of A.

Consider the determinant of , which is a polynomial of degree . The values of that solve the equation , are the eigenvalues of , and are as its degree.

We can write such a polynomial using its roots (its eigenvalues) as follows:

Its constant term is: ,

but it’s also the value of with , which obviously is .

So .

#### Trace of A.

If we develop the term of degree we obtain:

.

For simplicity we consider the case of an matrix:

It can be decomposed as follows:

Now we can apply the decomposition on the second row of the second matrix:

And finally we can apply the decomposition on the third row of the last matrix:

The matrices that give a contribute to the degree of are:

So their degree contribute is: .

Generalizing the contribution is: .

Finally