What does a Negative Determinant mean?

A negative determinant of a 2 x 2 matrix means that the column vectors have a clockwise orientation.

We first note that it is possible for a matrix to have a negative determinant.

The sign of the determinant gives us information about the orientation of the column vectors of the matrix.

For example, consider the following 2 x 2 matrix:

The determinant of the above matrix is det(A) = ad – bc = -2 -1 = -3.

Since the determinant is negative we conclude that the column vectors (-2, 1) and (1, 1) have a clockwise orientation.

This can be clearly seen by drawing the two vectors on a coordinate plane as shown below. Notice that the order in which the vectors occur is important when determining the orientation.

A positive determinant means that the 2 vectors have an anti-clockwise orientation. This orientation is also known as the standard orientation.

This is because the standard basis vectors e₁ = (1, 0) and e₂ = (0, 1) have an anti-clockwise orientation.

Notice that the matrix having e₁ = (1, 0) and e₂ = (0, 1) as column vectors is the identity matrix that has a positive determinant.

Example of a 3 x 3 matrix with negative determinant:

Consider the matrix:

The above matric has a determinant of -1. This is because the column vectors of the matrix do not have the standard orientation of e₁ = (1, 0,0), e₂ = (0, 1, 0), and e₃ = (0, 0, 1).

Instead, the column vectors occur in the order e₁, e₃, and e₂ which explains why the above matrix has a negative determinant.