Determinants are important when we want to understand how eigenvalues work in linear algebra.

• Eigenvalues can be thought of as numbers that show how a linear transformation changes vectors in a space. To put it simply, if we have a matrix called $A$, the eigenvalues, which we can call $\lambda$, come from a special equation called the characteristic polynomial: $\det(A - \lambda I) = 0$. Here, $I$ is just a special identity matrix.

• The determinant is useful because it gives us a geometric meaning: it measures the change in volume caused by the transformation represented by the matrix $A$. If the determinant is zero, it means that the transformation has squished some volume down to nothing. This happens along certain directions and tells us that $\lambda = 0$ is an eigenvalue.

• Each non-zero eigenvalue shows how much vectors get stretched or shrunk in the direction of certain eigenvectors. So, determinants help us find points where the transformation can’t be reversed, which is directly related to the eigenvalues. In other words,

$$\text{If } \det(A - \lambda I) = 0 \rightarrow \text{ eigenvalue } \lambda \text{ shows a change in volume.}$$

When we understand how determinants and eigenvalues are connected, we can see how linear transformations can change space. This shows why eigenvalues are so important when we look at how systems behave in linear algebra.