Overview:

Part 5: Determinants of 2 × 2 matrices

Calculating the determinant

The determinant of a 2 × 2 matrix M is written det M or |M|.

For a 2 × 2 matrix \(\begin{pmatrix} a & b\\c & d\end{pmatrix}\), the determinant can be written det\(\begin{pmatrix} a & b\\c & d\end{pmatrix}\) or \(\begin{vmatrix} a & b\\c & d\end{vmatrix}\) and is simply equal to \(ad – bc\).

Note: Determinants can only be found for square matrices. There is a general method for working out the determinant of an \(n \) × \(n \) matrix, described in Part 9 below. At that stage, you can check that the general method applied to a 2 × 2 matrix gives you the determinant \(ad – bc\).

What the determinant represents

The absolute value of the determinant of a 2 × 2 matrix M is equal to the area scale factor by which M transforms the areas of shapes. In particular, consider the parallelogram obtained by transforming the unit square. The unit square has area 1, so the parallelogram will have an area of |M|.

If the determinant is negative, it simply indicates a change of orientation. The vertices of the unit square are O, P, Q, and R going anticlockwise. If the vertices of the image O’, P’, Q’, and R’ also run anticlockwise, then the determinant is positive. If these vertices run clockwise i.e. the orientation has changed, this means that the determinant is negative.

A matrix with determinant of 0 is called a singular matrix.

Singular matrices map the unit square to a line—or to (0,0) in the case of the zero matrix. In these cases, the transformed “parallelograms” therefore have area 0.

A matrix with a non-zero determinant is called non-singular.