WOLFRAM|DEMONSTRATIONS PROJECT

3×3 Determinants by Expansion

​
expand along:
row
1
2
3
column
1
2
3
new numbers
Expand along row 1:
-1
2
3
4
-5
6
7
8
-9
=
- 1
-5
6
8
-9
- 2
4
6
7
-9
+ 3
4
-5
7
8
=
- 1(-3) - 2(-78) + 3(67) =
3 + 156 + 201 =
360
Consider the matrix:
M=
a
b
c
d
e
f
g
h
i
The determinant of
M
is the sum of three terms defined by a row or column. Each term is the product of an entry, a sign, and the minor for the entry. The signs look like this:
+
-
+
-
+
-
+
-
+
A minor is the 2×2 determinant formed by deleting the row and column for the entry. For example, this is the minor for the middle entry:

a
.
c
.
.
.
g
.
i
=
a
c
g
i
=ai-cg
Here is the expansion along the first row:
a
.
.
.
.
e
f
.
h
i

-b
.
.
.
d
.
f
g
.
i

+c
.
.
.
d
e
.
g
h
.

You would probably never write down the following matrix, but the patterns of the signs and the deleted rows and columns of the original matrix may be helpful. The determinant is the sum of any one of the rows or columns of this complicated matrix:
a
.
.
.
.
e
f
.
h
i

-b
.
.
.
d
.
f
g
.
i

c
.
.
.
d
e
.
g
h
.

-d
.
b
c
.
.
.
.
h
i

e
a
.
c
.
.
.
g
.
i

-f
a
b
.
.
.
.
g
h
.

g
.
b
c
.
e
f
.
.
.

-h
a
.
c
d
.
f
.
.
.

i
a
b
.
d
e
.
.
.
.
