More on Matrices - 3D Math Primer for Graphics and Game Development - page 162

a 2×2 or 3×3 matrix. Section 6.1.2 discusses minors and cofactors. Then,
Section 6.1.3 shows how to compute the determinant of an arbitrary n × n
matrix, by using minors and cofactors. Finally, Section 6.1.4 interprets the
determinant from a geometric perspective.

Determinants of 2 × 2 and 3 × 3 matrices

6.1.1

The determinant of a square matrix M is denoted | M | or, in some other
topics, as “det M .” The determinant of a nonsquare matrix is undefined.
This section shows how to compute determinants of 2×2 and 3×3 matrices.
The determinant of a general n × n matrix, which is fairly complicated, is
discussed in Section 6.1.3
The determinant of a 2 × 2 matrix is given by

Determinant of a 2 × 2
matrix

m 11 m 12
m 21 m 22

| M | = = m 11 m 22 − m 12 m 21 . (6.1)

Notice that when we write the determinant of a matrix, we replace the
brackets with vertical lines.
Equation (6.1) can be remembered easier with the following diagram.
Simply multiply entries along the diagonal and back-diagonal, then subtract
the back-diagonal term from the diagonal term.

Some examples help to clarify the simple calculation:

2 1

= (2)(2) − (1)(−1) = 4 + 1 = 5;

−1 2

−3

4

= (−3)(5) − (4)(2) = −15 − 8 = −23;

2 5

a b
c d

= ad − bc.