Function which describes the characteristics of a square matrix $A \in \R^{n \times n}$ in terms of the effect it has on the transformation of space.
$$ \det(A) = \begin{cases} 0 \hspace{36pt} \rightarrow A \text{ is not invertible} \\ \text{otherwise} \rightarrow A \text{ is invertible}\end{cases} $$
The absolute value of the determinant of a matrix is the hypervolume of the parallelepiped formed by the columns of the matrix.
Determinants can indicate whether or not a matrix is invertible. When a square, rank-deficient matrix transforms a vector space, each vector is mapped to a subspace of the domain. By the pigeonhole principle, there will exist at least one mapping which is not unique (i.e, the mapping is not one-to-one). Constructing an inverse to this map would imply that it is possible to reconstruct multiple vectors in the domain from a vector in the range, which is not feasible.
Given two vectors $\mathbf{u}, \mathbf{v} \in \R^{n}$, the determinant of $A$ describes the ratio between the hypervolume contained by $\mathbf{u}$ and $\mathbf{v}$ and the hypervolume contained between $A\mathbf{u}$ and $A\mathbf{v}$.
Note that when the columns of $A$ are linearly dependent, space is collapsed onto the span of its columns. Generally speaking, when a hypervolume loses one of its dimensions, its volume in its original space is said to have shrunk to zero.
$$ \det A = \frac{V'}{V} $$
When $\det A <0$, the orientation of the original space has “flipped”. In other words, a hypersurface subtended by the columns of $A$ oriented outward will now be oriented inward.
Values $\{\lambda _1 \dots \lambda_n \}$ which characterize the behavior of a linear transformation $A \in \R^{n \times n}$. Each distinct eigenvalue $\lambda _i$ is associated with an eigenvector $\mathbf{v}_i$ which, upon being transformed by $A$, is simply just scaled by $\lambda _i$.
$$ A\mathbf{v}_i = \lambda_i \mathbf{v}_i \hspace{15pt} \text{ for } i \in \{1 \dots n\} $$
The eigenvalues of $A$ are found through the roots of its characteristic polynomial. Since the degree of this polynomial is $n$, we are guaranteed $n$ roots by the Fundamental Theorem of Algebra.
$$ \det A- \lambda I = 0 \rightarrow p(\lambda) = 0 \hspace{15pt} \text{ with } \deg(p) = n $$
A solution $\lambda _i$ is characterized by its algebraic multiplicity, the number of times it appears as a root, and its geometric multiplicity, the dimension of the space spanned by its associated eigenvector(s).