Determinant

Definition

Let $K$ be a commutative ring and let $n \ge 1$. There is exactly one map $\det \colon M_n(K) \to K$ with these properties:

1. $\det A$ is a multilinear function of the $n$ rows of $A$.
2. $\det A = 0$ when two rows of $A$ are equal.
3. $\det I_n = 1$, for the identity matrix $I_n$.

This map is the determinant.

Remarks

In other words, $\det$ is the unique alternating multilinear map of the rows with value $1$ on $I_n$. The Leibniz formula gives $\det A$ as a sum over permutations. For $n = 2$, $\det \begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc$. The same characterization holds with columns in place of rows, because $\det A^T = \det A$.

Explanations

L1 human

The determinant gives one number for each square matrix. Over the real numbers, its absolute value is the factor by which the matrix scales volume, and its sign tells if the matrix flips orientation. A real square matrix is invertible exactly when its determinant is not zero.