Determinants Seen Geometrically

dimension

2D

3D

2D

a

1.

b

0.

c

0.

d

1.

3D

a

1

b

0

c

0

d

0

e

1

f

0

g

0

h

0

i

1

zoom

2



a	b
c	d

 = 

1.00	0.00
0.00	1.00

 = 1.

The determinant of a

2×2

matrix



a	b
c	d



is the area of the parallelogram with the column vectors



a

c



and



b

d



as two of its sides.

Similarly, the determinant of a

3×3

matrix

a	b	c
d	e	f
g	h	i

is the volume of the parallelepiped (skew box) with the column vectors

a

d

g

,

b

e

h

, and

c

f

i

as three of its edges.

Color indicates sign.

When the column vectors are linearly dependent, the parallelogram or parallelepiped flattens down at least one dimension and area or volume is zero. Other determinant facts have corresponding geometric interpretations. For instance, doubling a column doubles the area or volume because that doubles one of the dimensions of the parallelogram or parallelepiped.