Determinants
Determinant - a square array of
numbers or variables enclosed between
parallel vertical bars.
**To find a determinant you must have
a SQUARE MATRIX!!**
Finding a 2 x 2 determinant:
a
b
c d
= ad - bc
Find the determinant:
1.
5 7
11
8

(-5)(8) – (-7)(11)
(-40) – (-77)
37
2.
3
2
1 5

(3)(5) – (2)(-1)
(15) – (-2)
17
Challenge:
(-4)(x) – (1)(3) = 17
-4x – 3 = 17
-4x = 20
x = -5
3x3 Determinant: Diagonal method
4.
4.
2
3
6
7 1
4
5
9
2
3
8 2
3
6
7 1 6
7
4
5
9 4
5
8
Step 1: Rewrite first two
columns of the matrix.
-224 +10 +162 = -52
4.
2
3
8 2
3
6
7 1 6
7
4
5
9 4
5
Step 2: multiply
diagonals going down!
Step 3: multiply
diagonals going up!
-126 +12 +240 =126
126 - (-52)
= 178
Last Step:
(step 2) – (step 3)
Now you try:
-18 +50 +6 = 38
5 1
2
5
1
5. 2 3
5
2
3
3 3
2
3
2
38 - 38
=0
45 - 15 + 8 = 38
3x3 Determinant: Expansion by Minors method
Imagine crossing out the first row. Now take the double-crossed
element. . .
And the first column.
And multiply it by the
determinant of the remaining
2x2 matrix
3
8
¼
2
0
-¾
4
180
11
3x3 Determinant: Expansion by Minors method
Keep first row crossed out
Now cross out the second column
3
8
¼
2
0
-¾
4
180
11
Now take the negative of the
double-crossed element. . .
And multiply it by the
determinant of the remaining
2x2 matrix
3x3 Determinant: Expansion by Minors method
Finally, cross out first
row and last column
3
8
¼
2
0
-¾
4
180
11
Use double-crossed element
And multiply it by the
determinant of the remaining
2x2 matrix
Now you try:
5 1
2
5. 2 3
5
3
3
2
5[9-10]
- (-1) [-6 -15]
5(-1)
1 (-21)
-5
-21
2[4- (-9)]
2(13)
26
Use determinant to find area of a triangle
Find the area of a triangle whose vertices
are located at (–3, –3), (–1, 2), and (3, –1).
A=
1
=
=
2
1
2
1
2
a
b 1
c
d 1
e
f
1
–3 –3 1
–1
2
3
–1 1
1
[–26] or –13
13 units2
(a, b) = (–3, –3)
(c, d) = (–1, 2)
(e, f) = (3, –1)
Simplify.
*Remember area cannot be
negative, thus we take the
absolute value