Determinant

The determinant is a function from square matrices to the underlying field. It is the unique alternating multilinear function of the columns that sends the identity matrix to $1$.

The determinant encodes invertibility: a matrix $A$ is invertible if and only if $\det(A) \neq 0$. It is multiplicative: $\det(AB) = \det(A)\det(B)$. Geometrically, $|\det(A)|$ measures the factor by which the linear map scales volumes, and the sign records whether orientation is preserved or reversed.

For a $2 \times 2$ matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$, the determinant is $ad - bc$.