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.
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.
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.
