Math 314/814 Matrix Theory
Exam 2 Practice problems
Show all work. Include all steps necessary to arrive at an answer unaided by a mechanical
computational device. The steps you take to your answer are just as important, if not more
important, than the answer itself. If you think it, write it!
1. The company that markets Brand X toothpaste plans to introduce their product in a
town where everyone uses Brand Z. Their research indicates that during each month, 2/5
of the brand X users will switch to their brand (the remainder will remain loyal), while
7/10ths of the users of their brand will switch back to Brand X. What is the transition
matrix that describes this Markov process?
Based on this research, what fraction of the town’s population will be Brand Z users
after 3 months?
2. Find a basis for R3from among the vectors
1
2
3
,
3
2
1
,
1
1
1
,
−1
2
1
, which includes the vector
1
1
1
.
3. Suppose T:R3→R3is a linear transformation, and suppose that
T
1
0
0
=
−1
2
3
,T
0
1
0
=
3
−2
1
, and T
0
0
1
=
1
−3
5
.
What is T
2
3
−5
?
4. Find the determinant of the matrix B=
1 3 2 2
2 5 2 2
0 2 3 4
3 3 1 3
.
5. The matrix A=
3−8 3
2−14 6
2−31 14
has characteristic polynomial
χA(λ) = −(λ+ 1)(λ−2)2.
Find the eigenvalues of Aand bases for each of the corresponding eigenspaces.
Is Adiagonalizable? Why or why not?
6. Let V=R3(3-dimensional Euclidean space) and let
W = {(x, y, z)∈R3: 2x+ 3y+ 8z= 0}.
Show that W is a subspace of V.
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