Determinant

A scalar invariant of a square matrix measuring volume scaling and invertibility.

Determinant

A determinant is a function that assigns to each $n\times n$ matrix $A=(a_{ij})$ over a field $\mathbb{F}$ the scalar

\det(A)=\sum_{\sigma\in S_n}\operatorname{sgn}(\sigma)\prod_{i=1}^n a_{i,\sigma(i)},

where $S_n$ is the set of permutations of $\{1,\dots,n\}$ and $\operatorname{sgn}(\sigma)\in\{\pm 1\}$ is the sign of $\sigma$ .

For a linear operator $T:V\to V$ on a finite-dimensional vector space , one defines $\det(T)$ as the determinant of any matrix representing $T$ in a basis (this does not depend on the choice of basis). The characteristic polynomial is defined using determinants.

Examples:

If $A$ is diagonal with diagonal entries $d_1,\dots,d_n$ , then $\det(A)=d_1\cdots d_n$ .
If $A$ is upper triangular, then $\det(A)$ is the product of its diagonal entries.
For the scaling operator $T(v)=c\,v$ on an $n$ -dimensional space, $\det(T)=c^n$ .