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What does a Negative Determinant mean?

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

For example, consider the following 2 x 2 matrix, A= \begin{bmatrix} -2 & 1 \\ 1 & 1 \end{bmatrix}. 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 co-ordinate plane as shown below. Notice that the order in which the vectors occur is important when determining the orientation.

Two vectors oriented clockwise having negative determinant
Two vectors oriented clockwise having negative determinant

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 e_1 = (1, 0) and e_2 = (0, 1) have an anti-clockwise orientation.

Standard basis vectors have anti-clockwise orientation and positive determinant
Standard basis vectors have anti-clockwise orientation and positive determinant

Notice that the matrix having e_1 = (1, 0) and e_2 = (0, 1) as column vectors is the identity matrix which has a positive determinant.

Example of a 3 x 3 matrix with negative determinant:

Consider the matrix A= \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix}. The above matric has a determinant of -1. This is because the column vectors of the matrix do not have the standard orientation of e_1 = (1, 0, 0), e_2 = (0, 1, 0) and e_3 = (0, 0, 1). Instead the column vectors occur in the order e_1 , e_3 \text{ and } e_2 which explains why the above matrix has a negative determinant.

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