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.