Determinants
Every square matrix, A, is assigned a particular scalar quantity called the determinant of A, denoted by |A|or by det (A).
A = 
Determinant of Order One
|a11| = a11
|5| = 5
Determinant of Order Two
= a 11 a 22 − a 12 a 21
Determinant of Order Three
Consider an arbitrary 3 x 3 matrix, A = (aij). The determinant of A is defined as follows:
=
a11 a22 a33 + a12 a23 a 31 + a13 a21 a32 -
- a 13 a22 a31 - a12 a21 a 33 - a11 a23 a32.
=
3 · 2 · 4 + 2 · (-5) · (-2) + 1 · 0 · 1 -
- 1 · 2 · (-2) - 2 · 0 · 4 - 3 · (-5) · 1 =
= 24 + 20 + 0 - (-4) - 0 - (-15) =
= 44 + 4 + 15 = 63
Note that there are six products, each consisting of three elements in the matrix. Three of the products appear with a positive sign (they preserve their sign) and three with a negative sign (they change their sign).
