The determinant of a matrix is really important for understanding its eigenvalues. Let's break this down into simpler parts.

1. What Are Eigenvalues?
   For a matrix called $A$, we can find its eigenvalues (which we label as $\lambda$) by using a special equation. This equation is written like this:
   ( det(A - \lambda I) = 0 )
   Here, ( I ) is a special matrix known as the identity matrix.

2. Types of Eigenvalues:

   • When the Determinant is Not Zero (( \det(A) \neq 0 )):
     This means all eigenvalues are different and none of them are zero. This happens when our main equation has a certain number of terms, which is called the degree ( n ).

   • When the Determinant is Zero (( \det(A) = 0 )):
     This tells us that at least one eigenvalue is zero. This can lead to some confusion in the matrix, showing that some rows or columns depend on each other.

3. Multiplicity of Eigenvalues:
   The number of times an eigenvalue appears is called its multiplicity. We can find this by looking at how eigenvalues show up in our main equation.

So, to sum it up:
The determinant is a really key factor in figuring out if eigenvalues are there and what type they are when we look at a matrix.