determinant

A single number computed from a square matrix that reveals whether the system it represents has a unique solution.

The determinant of a $2\times 2$ matrix $\begin{pmatrix}a&b\\c&d\end{pmatrix}$ is $ad-bc$ ; larger square matrices expand by cofactors along a row or column into smaller determinants. A determinant of zero means the matrix's rows are proportional — geometrically, the parallelogram or box they span has been squashed flat — and signals no unique solution and no matrix inverse.