determinant

A single number, $\det A$ , reporting a matrix's signed scaling factor.

For $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$ , $\det A = ad-bc$ — the signed area of the parallelogram swept out by $A$ 's columns, positive when the second column sits counterclockwise from the first, negative when clockwise. For larger matrices $\det A$ is computed by cofactor expansion, and it detects invertibility: $\det A \ne 0$ exactly when the square matrix $A$ is invertible.