The determinant of a matrix $M$ is written as either $|M|$ or $\det($M).$

The determinant of a 2x2 matrix is the area scale-factor.
The determinant of a 3x3 matrix is the volume scale-factor.

In general, when a matrix is thought of as a transformation, the determinant gives the overall amount of stretching of squeezing. A determinant of 1 means that area or volume (or more generally, "measure") is preserved. A determinant of 0 means that the unit object ends up with no area or volume (or more generally, "measure"), and therefore the transformation cannot be undone.

The determinant of a 2x2 matrix $\left[\begin{matrix}a&b\\c&d\end{matrix}\right]$ is $\left|\begin{matrix}a&b\\c&d\end{matrix}\right|~=~ad-bc$

Determinants can be used to find the 2D vector cross product.

See also Matrices

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Determinant Of A Matrix

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