# 2x2 Determinant

The determinant of matrix $$M$$ is commonly denoted as $$det(M)$$, $$det M$$ or $$|M|$$. In practice, the absolute value of the determinant gives the scale factor by which the matrix will scale a volume. This definition is not really used in games, for games the determinant is a helper function to find the inverse of a matrix.

The determinant of a matrix is the same as the determinant of its transpose, $$|M| = |M^{T}|$$. Similarly, the determinant of the inverse of a matrix is the same as the inverse of it's determinant: $$|M^{-1}| = \frac{1}{|M|}$$

Finding the determinant of a matrix is actually a recursive operation. Before learning about the recursive nature of the operation, let's explore the determinant of our smallest square matrix, a 2x2 matrix. Consider this 2x2 matrix:

$$M = \begin{bmatrix} a & c\\ b & d \end{bmatrix}$$

To find the determinant of the above matrix, you need to multiply diagonal elements and subtract the result. For example, the determinant of the above matrix would be a * d - c * b. A simple way to express this is:

$$|M| = | \begin{bmatrix} a & c\\ b & d \end{bmatrix} | = ad - cb$$

float Determinant(mat2 m) {
return m.v[0] * m.v[3] - m.v[2] * m.v[1];
}