Any square matrix has a determinant—an important number associated with that matrix. Nonsquare matrices do not have a determinant.
How do you find the determinant of a 3x3 matrix? The method presented here is referred to as “expansion by minors.” There are other methods, but they turn out to be mathematically equivalent to this one: that is, they end up doing the same arithmetic and arriving at the same answer.
Table 1
Example: Finding the Determinant of a 3x3 Matrix 
Find the determinant of
24510831112451083111 size 12{ left [ matrix {
2 {} # 4 {} # 5 {} ##
"10" {} # 8 {} # 3 {} ##
1 {} # 1 {} # 1{}
} right ]} {} 
The problem.

***SORRY, THIS MEDIA TYPE IS NOT SUPPORTED.***

We’re going to walk through the top row, one element at a time, starting with the first element (the 2). In each case, begin by crossing out the row and column that contain that number.

∣8311∣∣8311∣ size 12{ lline matrix {
8 {} # 3 {} ##
1 {} # 1{}
} rline } {}=(8)(1)–(3)(1)=5 
Once you cross out one row and column, you are left with a 2x2 matrix (a “minor”). Take the determinant of that matrix.

2(5)=10 
Now, that “minor” is what we got by crossing out a 2 in the top row. Multiply that number in the top row (2) by the determinant of the minor (5).

***SORRY, THIS MEDIA TYPE IS NOT SUPPORTED.***
(10)(1)–(3)(1)= 74(7)=28 
Same operation for the second element in the row (the 4 in this case)...

***SORRY, THIS MEDIA TYPE IS NOT SUPPORTED.***
(10)(1)–(8)(1) = 25(2)=10 
...and the third (the 5 in this case).

+10 – 28 + 10 = –8 
Take these numbers, and alternately add and subtract them; add the first, subtract the second, add the third. The result of all that is the determinant.

This entire process can be written more concisely as:

2
4
5
10
8
3
1
1
1

=
2

8
3
1
1

4

10
3
1
1

+
5

10
8
1
1

=
2
(
5
)

4
(
7
)
+
5
(
2
)
=
8

2
4
5
10
8
3
1
1
1
=2
8
3
1
1
4
10
3
1
1
+5
10
8
1
1
=2(5)4(7)+5(2)=8
This method of “expansion of minors” can be extended upward to any higherorder square matrix. For instance, for a 4x4 matrix, each “minor” that is left when you cross out a row and column is a 3x3 matrix. To find the determinant of the 4x4, you have to find the determinants of all four 3x3 minors!
Fortunately, your calculator can also find determinants. Enter the matrix given above as matrix [D]. Then type:
MATRX
► 1
MATRX 4 ) ENTER
The screen should now look like this:
If you watched the calculator during that sequence, you saw that the rightarrow key took you to the MATH
submenu within the MATRIX
menus. The first item in that submenu is DET (
which means “determinant of.”
What does the determinant mean? It turns out that this particular odd set of operations has a surprising number of applications. We have already seen one—in the case of a 2x2 matrix, the determinant is part of the inverse. And for any square matrix, the determinant tells you whether the matrix has an inverse at all.
Another application is for finding the area of triangles. To find the area of a triangle whose vertices are (a,b), (c,d), and (e,f), you can use the formula: Area = ½
∣acebdf111∣∣acebdf111∣ size 12{ lline matrix {
a {} # c {} # e {} ##
b {} # d {} # f {} ##
1 {} # 1 {} # 1{}
} rline } {}. Hence, if you draw a triangle with vertices (2,10), (4,8), and (5,3), the above calculation shows that the area of this triangle will be 4.
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