1. Lesson 6
Introduction to Determinants (Section 13.1–2)
Math 20
October 1, 2007
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Thomas Schelling at IOP (79 JFK Street), Wednesday 6pm
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3. G , A , and the Euclidean algorithm
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4. Consider the system of two equations in two variables:
a11 x1 +a12 x2 =b1
a21 x1 +a22 x2 =b2
Can you find the solutions for x1 and x2 in terms of the
coefficients?

6. Solutions
The solutions are
a22 b1 − a12 b2
x1 =
a11 a22 − a21 a12
a11 b2 − a21 b1
x2 =
a11 a22 − a21 a12

8. Solutions
The solutions are
a22 b1 − a12 b2
x1 =
a11 a22 − a21 a12
a11 b2 − a21 b1
x2 =
a11 a22 − a21 a12
Observations
If a11 a22 − a21 a12 = 0, the system has a unique solution.
If a11 a22 − a21 a12 = 0 and a22 b1 − a12 b2 = 0, the system has
no solution
If a11 a22 − a21 a12 = 0 and a22 b1 − a12 b2 = 0, the system has
infinitely many solutions

9. The determinant
Definition
a11 a12
The determinant of a 2 × 2 matrix A = is the number
a21 a22
a11 a12
= a11 a22 − a21 a12
a21 a22

13. Theorem (Cramer’s rule)
The solution to the system of linear equations
a11 x1 +a12 x2 =b1
a21 x1 +a22 x2 =b2
is
b1 a12
a22 b1 − a12 b2 b2 a22
x1 = =
a11 a22 − a21 a12 a11 a12
a21 a22
a11 b1
a11 b2 − a21 b1 a21 b2
x2 = =
a11 a22 − a21 a12 a11 a12
a21 a22

14. Leontief
Example
On the first day we had to solve the system
0.6x1 − 0.3x2 = 75000
−0.2x1 + 0.7x2 = 50000

16. Leontief
Example
On the first day we had to solve the system
0.6x1 − 0.3x2 = 75000
−0.2x1 + 0.7x2 = 50000
Solution
67, 500
x1 = = 187, 500
0.36
45, 000
x2 = = 125, 000
0.36

20. The 3 × 3 case
Does anybody want to do:
a11 x1 +a12 x2 +a13 x3 =b1
a21 x1 +a22 x2 +a23 x3 =b2
a31 x1 +a32 x2 +a33 x3 =b3

21. The 3 × 3 case
Does anybody want to do:
a11 x1 +a12 x2 +a13 x3 =b1
a21 x1 +a22 x2 +a23 x3 =b2
a31 x1 +a32 x2 +a33 x3 =b3
We get
b1 a22 a33 − b1 a23 a32 − b2 a12 a33
+ b2 a13 a32 + b3 a12 a23 − b3 a22 a13
x1 =
a11 a22 a33 − a11 a23 a32 − a21 a12 a33
+ a21 a13 a32 + a31 a12 a23 − a31 a22 a13

22. Definition
The determinant of a 3 × 3 matrix is
a11 a12 a13
a21 a22 a23 = a11 a22 a33 − a11 a23 a32 − a21 a12 a33
a31 a32 a33
+ a21 a13 a32 + a31 a12 a23 − a31 a22 a13

23. Cofactors
We can compute a 3 × 3 determinant in terms of smaller
determinants:
a11 a12 a13
a a a a a a
a21 a22 a23 = a11 22 23 − a12 21 23 + a13 21 22
a32 a33 a31 a33 a31 a32
a31 a32 a33

24. Theorem (Cramer’s rule)
The solution to the 3 × 3 system
a11 x1 +a12 x2 +a13 x3 =b1
a21 x1 +a22 x2 +a23 x3 =b2
a31 x1 +a32 x2 +a33 x3 =b3
is
b1 a12 a13 a11 b1 a13 a11 a12 b1
b2 a22 a23 a21 b2 a23 a21 a22 b2
b3 a32 a33 a31 b3 a33 a31 a32 b3
x1 = , x2 = , x3 =
a11 a12 a13 a11 a12 a13 a11 a12 a13
a21 a22 a23 a21 a22 a23 a21 a22 a23
a31 a32 a33 a31 a32 a33 a31 a32 a33

25. Example
Solve
x1 +2x2 +3x3 = 6
2x1 −3x2 +2x3 = 14
3x1 + x2 − x3 =−2

26. Example
Solve
x1 +2x2 +3x3 = 6
2x1 −3x2 +2x3 = 14
3x1 + x2 − x3 =−2
Solution
50 −100 150
x1 = 50 = 1, x2 = 50 = −2, x3 = 50 =3

27. A geometric interpretation