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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