Math 2331 Linear Algebra

Test 1 - Chapters 1 and 2 Practice Problems

Use the following vectors for questions 1-7

12

42

u

⎛ ⎞⎜ ⎟−⎜=⎜⎜ ⎟⎝ ⎠

⎟⎟

⎟⎟

1010

v

−⎛ ⎞⎜ ⎟⎜=⎜⎜ ⎟⎝ ⎠

12

121212

w

⎛ ⎞−⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟

= ⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟−⎜ ⎟⎝ ⎠

1. Find u wi 2. Find u vi 3. Find 2 4u w− + 4. Find v w− − 5. Are v and w orthogonal?

6. Find the length of each vector. 7. State a unit vector in the direction of u. Answer questions 8-12 for each of the following systems:

3 42 4

x yx y− =

− − = 2

15

x yx y− = −+ =

5 53 5

x yx y+ =

− + = −

8. Sketch the ROW PICTURE of the solution to the system. 9. Sketch the COLUMN PICTURE of the solution to the system. 10. State the system as a matrix equation. 11. State the elimination matrix E that transforms the system to upper triangular. 12. Solve by elimination and back substitution.

3 42 4

x yx y− =

− − = 2

15

x yx y− = −+ =

5 53 5

x yx y+ =

− + = −

State the LU and LDU factorization of each of the following matrices.

13. 1 34 5

A ⎛ ⎞= ⎜ ⎟⎝ ⎠

14. 1 2 32 3 03 0 4

B⎛ ⎞⎜ ⎟= ⎜ ⎟⎜ ⎟⎝ ⎠

15.

2 1 0 01 2 1 00 1 2 10 0 1 2

C

⎛ ⎞⎜ ⎟⎜ ⎟=⎜ ⎟⎜ ⎟⎝ ⎠

Find the inverse, if it exists, of each of the following matrices.

16. 1 1 34 0 52 1 3

A−⎛ ⎞

⎜ ⎟= ⎜ ⎟⎜ ⎟− −⎝ ⎠

17. 4 92 4

M−⎛ ⎞

= ⎜ ⎟−⎝ ⎠

18. 1 43 12

B ⎛ ⎞= ⎜ ⎟⎝ ⎠

19. 2 1 01 2 10 1 2

A⎛ ⎞⎜ ⎟= ⎜ ⎟⎜ ⎟⎝ ⎠

20. State the transpose of matrices in problems 16-19

21. Given the matrices in problems 13-19, which are symmetric? 22. State the 4x4 permutation matrix P that switches row 1 and row 3. What is the inverse of P? Given the following matrices and vectors,

1 1 34 0 52 1 3

A−⎛ ⎞

⎜ ⎟= ⎜ ⎟⎜ ⎟− −⎝ ⎠

1 2 34 1 7

M ⎛ ⎞= ⎜ ⎟−⎝ ⎠

0 1 01 0 00 0 1

P⎛ ⎞⎜ ⎟= ⎜ ⎟⎜ ⎟⎝ ⎠

1 32 1

0 5Q

⎛ ⎞⎜ ⎟= −⎜ ⎟⎜ ⎟⎝ ⎠

23

0b

⎛ ⎞⎜ ⎟= −⎜ ⎟⎜ ⎟⎝ ⎠

54

c ⎛ ⎞= ⎜ ⎟⎝ ⎠

111

x⎛ ⎞⎜ ⎟= ⎜ ⎟⎜ ⎟⎝ ⎠

23. Multiply Ax = Ab = 24. Multiply Mc Mb

25. Multiply AM MA 26. Multiply QM PQ 27. Find the transpose of A, M and b 28. Find the transpose of PQ and MP

Solve the following systems by elimination and back substitution:

29.

3 2 0 0 14 5 0 0 30 0 6 5 40 0 7 6 1

xyzt

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟ =⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

30. 2 2

4 8 93 2

x y zx y z

y z

+ + =+ + =

+ =

131