Determinant of 2 by 2 Matrix Formula
The determinant of a 2x2 matrix returns a number that measures the change in area of the matrix transformation. For example, the determinant of the matrix is .
When we visualize the transformation represented by the matrix, the area between the basis vectors starts as one unit squared and is transformed to the parallelogram whose area is equal to units squared.
When the orientation of the basis vectors gets flipped during the transformation, the determinant returns a negative number. Take, for example, the matrix with the columns are and whose determinant is .
When we play the matrix’s transformation, the area between the basis vectors grows by a factor of and their orientation changes.
The formula for the determinant of a three by three matrix.