thesaurus.maths.org

The determinant of a matrix is a number which can tell us something about the properties of the matrix. If the matrix represents a 2D transformation, then the determinant will tell us the ratio by which areas are changed, for example.
Often the determinant of (

a	b
c	d

) is written |

a	b
c	d

|
For a 2 ×2 matrix, A = (

a	b
c	d

) ,
the determinant is:

det(A) = ad − bc

In this case the determinant is the area of a parallelogram with sides given by the vectors (

a

c

), (

b

d

).
For a 3 ×3 matrix, we can find the determinant as follows:

     a b c d e f g h i      = a ×     e f h i     − b ×     d f g i     + c ×     d e g h    

In this case it is the volume of the parallelepiped with edges given by the vectors (

a

d

g

), (

b

e

h

) , (

c

f

i

) .