1. APPLICATIONS EIGENVALUES & EIGENVECTORS
CHAPTER 2 MATRICES
APPLICATIONS
EIGENVALUES & EIGENVECTORS

2. EIGENVALUES & EIGENVECTORS
Definition 2.26 Consider the nxn matrix A. The eigenvalue, ʎ and the eigenvector X are defined by: A X= ʎ X Such that X is said to be an eigenvector corresponding to eigenvalue ,ʎ .

3. EIGENVALUES & EIGENVECTORS
STEP OF SOLUTIONS
compute the determinant of
Find the roots of the polynomial:
For each eigenvalue, : Find eigenvectors by solving

4. EIGENVALUES & EIGENVECTORS
Example 1 Consider matrix . Find the eigenvalues and eigenvectors for matrix.

5. EIGENVALUES & EIGENVECTORS
Solution Step 1: Compute

6. EIGENVALUES & EIGENVECTORS
Solution Therefore:

7. EIGENVALUES & EIGENVECTORS
Solution Step 2: Find the roots (ʎ)

8. EIGENVALUES & EIGENVECTORS
Solution Step 3: Find the eigenvectors for each eigenvalue : When

9. EIGENVALUES & EIGENVECTORS
Solution Therefore : Solving eigenvector, X:

10. EIGENVALUES & EIGENVECTORS
Solution Reduce using Gauss elimination:

11. EIGENVALUES & EIGENVECTORS
Solution Therefore, solve from 2nd row : Let and, From 1st row:

12. EIGENVALUES & EIGENVECTORS
Solution Therefore : The eigenvector for

13. EIGENVALUES & EIGENVECTORS
Solution Step 3: Find the eigenvectors for each eigenvalue : When

14. EIGENVALUES & EIGENVECTORS
Solution Therefore : Solving eigenvector, X:

16. EIGENVALUES & EIGENVECTORS
Solution Therefore, from 2nd row : From 1st row: Since the coefficient of , then : Let The eigenvector for

17. EIGENVALUES & EIGENVECTORS
Solution Step 3: Find the eigenvectors for each eigenvalue : When

18. EIGENVALUES & EIGENVECTORS
Solution Therefore : Solving eigenvector, X:

19. EIGENVALUES & EIGENVECTORS
Solution Reduce using Gauss elimination:

20. EIGENVALUES & EIGENVECTORS
Solution Therefore from 2nd row : let 1st row:

21. EIGENVALUES & EIGENVECTORS
The eigenvector for

22. EIGENVALUES & EIGENVECTORS
Example 2
Consider matrix
Find the eigenvalues and eigenvectors for matrix A using Gauss Elimination method.

23. EIGENVALUES & EIGENVECTORS
Solution Step 1: Compute

24. EIGENVALUES & EIGENVECTORS
Solution Therefore:

25. EIGENVALUES & EIGENVECTORS
Solution Step 2: Find the roots (ʎ) :

26. EIGENVALUES & EIGENVECTORS
Solution Step 3: Find the eigenvectors for each eigenvalue : When

27. EIGENVALUES & EIGENVECTORS
Solution Therefore : Solving eigenvector, X:

28. EIGENVALUES & EIGENVECTORS
Solution Reduce using Gauss elimination:

29. EIGENVALUES & EIGENVECTORS
Solution Therefore from 1st row: Let , and Then The eigenvector for

30. EIGENVALUES & EIGENVECTORS
Solution Step 3: Find the eigenvectors for each eigenvalue : When

31. EIGENVALUES & EIGENVECTORS
Solution Therefore : Solving eigenvector, X:

32. EIGENVALUES & EIGENVECTORS
Solution Reduce using Gauss elimination:

33. EIGENVALUES & EIGENVECTORS
Solution
Reduce using Gauss elimination:

34. EIGENVALUES & EIGENVECTORS
Solution Therefore from 2nd row: , let 1st row:

35. EIGENVALUES & EIGENVECTORS
The eigenvector for

36. APPLICATIONS OF MATRICES
Knowing that the equation of a circle may be written in the form find the equation of the circle passing through the points (-3,-7), (4, -8) and (1,1)

37. Solution

38. Solution
Systems of linear equations:

39. Solution
Using Gauss Elimination:

40. Solution
Using Gauss Elimination:
Therefore, using backward substitution:

41. APPLICATIONS OF MATRICES
A cup of uncooked rice contains 15g of protein and 810 calories. A cup of uncooked soybeans contains 22.5g of protein and 270 calories. How many cups of each should be used for a meal containing 9.5g of protein and 324 calories? (Ans: 1/3 cup of uncooked rice and 1/5 cup of uncooked soybeans)

42. Solution Systems of linear eq: RICE (R) SOYBEANS (S) TOTAL PROTEIN (g)15
22.5
9.5
CALORIES
810
270
324

43. Solution
1/3 cup of uncooked rice and 1/5 cup of uncooked soybeans

44. APPLICATIONS OF MATRICES
The Waputi Indians make woven blankets, rugs and skirts. Each blanket requires 24 hours for spinning the yarn, 4 hours for dyeing the yarn and15 hours for weaving. Rugs require 30,5 and 18 hours and skirts 12,3 and 9 hours, respectively. If there are 306, 59 and 201 hours available for spinning, dyeing and weaving, respectively, how many of each item can be made? (Ans: 5 blankets, 3 rugs and 8 skirts)

45. APPLICATIONS OF MATRICES
A supplier of agricultural products has three types of fruit tress fertilizer, X, Y and Z, having nitrogen contents of 40%, 30% and 10%, respectively. The supplier plans to mix them, obtaining 700kg of fertilizer with a 25% nitrogen content. The mixture should contain 150kg more of type Z, than type Y. How much of each type should be used? Solve this problem using Cramer's Rule. (Ans: X=250, Y=150 and Z=300)