Determinant of 3 by 3 Matrix Formula
The determinant of a 3x3 matrix returns a number that measures the change in volume of the matrix transformation. For example, the determinant of the matrix is .
When we play the matrix’s transformation, the volume of the parallelepiped formed between the basis vectors is scaled by a factor of .
The determinant returns both positive and negative values. When the value is positive, the transformed orientation of the basis vectors is the same as their starting orientation. When the value is negative, the orientation of the space has been inverted. We can demonstrate this by flipping the signs of the values of one of the columns in the matrix.
For example, if we flip the signs of the values in the second column of the matrix, space is still transformed by the same amount, but the determinant’s value is now negative.
When we play the new matrix’s transformation, the basis vector crosses the plane formed by the other two basis vectors and inverts the orientation of space.
The determinant of a 2x2 matrix returns a number that represents the change in area of the matrix transformation.