# PracticeProblems3Fall2012.pdf

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Georgia Institute of Technology

Mathematics

MATH 1502

Blekherman

Fall

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Practice Problem for Midterm #3, Math 1502
1. Find and simplify the following determinants:
▯ ▯ ▯ ▯ ▯ ▯
▯ -1 0 5 ▯ ▯ ▯1 b ▯ ▯a 1 2 ▯
a)▯ 1 -2 -1 ▯ b)▯ ▯ c)▯a 2 ▯1 ▯
▯ 3 -1 2 ▯ x y ▯a ▯1 2 ▯
▯ ▯ ▯-1 3 3 4 ▯
▯ ▯1 ▯ ▯ 0 5 ▯ ▯ ▯
d)▯ 1 ▯2 ▯ ▯ ▯1 ▯ e) ▯0 -2 1 3 ▯
▯ ▯ ▯0 0 3 4 ▯
3 ▯1 2 ▯ ▯ ▯0 0 0 3 ▯
2. Find the characteristic polynomial, eigenvalues and eigenvectors of the following matrices:
▯ ▯ ▯ ▯ ▯ ▯
-2 4 2 1 4 1
a) 9 -7 b) 12 1 c) 3 2
0 1
-1 3 3 4 0 1 0 1
B 0 -2 1 3 C 1 0 1 1 1 1
d)@ A e)@ 0 1 ▯1 A f)@ 2 2 2 A
0 0 3 4 5 ▯1 2 3 3 3
0 0 0 3
3. Find the inverses of the following matrices, if possible:
0 1 0 1
▯ ▯ ▯ ▯ 1 1 5 2 1 -1
1 3 1 4 @ A @ A
a) 2 -6 b) 2 8 c) 3 2 1 d) 1 2 1
2 1 -2 -3 0 -2
4. Find the basis of the kernel and the column space of the following matrices and determine their
rank.
0 1
▯ ▯ 0 1 4 1 0 1 3 1
1 3 2 -3 1 B -1 2 C
a) 2 6 1 -2 4 b) @ 2 -4 3 -1 A c)@ 2 1 A
-2 3 -5 2
-5 6
5. Find the dimension and a basis of the following subspaces:
0 1 1 0 4 1 0 3 1
a) The span of 6 A , -1 A and @ -4 A.
-2 3 4
0 1
x1
B C
b) All vectors x2C in R with 1 + 2 + x3+ x4= 0. (Hint: Think of this as a kernel of some
@ x3A
x4
matrix A).
1 0 1
x1 3
c) All vectorsx2 A in R with 1 + 2 ▯ 33 = 0 and 1 ▯ 2 + 23 = 0 (Hint: Think of this as
x3
a kernel of some matrix A).
6. Diagonalize the following matrices, if possible:
0 1 0 1 0 1
▯ ▯ 1 4 1 2 0 1 2 0 0 0
1 -2 B 0 1 2 0 C
a) 2 6 b)@ 0 -4 3 A c)@ 0 1 0 A d)@ 0 0 -5 0 A
0 0 2 1 0 2
0 0 0 -3
7. True or False. No partial credit.
(a) A singular matrix always has deter

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