MATH 221 - EXAM #2
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Student ID:
Exam rules:
• No calculators, open books or notes are allowed.
• There are 9 problems in this exam. The number of points for each problem
is given in the brackets.
1
Problem 1. [5 Pts] Find the eigenvalues of the matrix below. For each eigenvalue
indicate its algebraic multiplicity.


1 0 2 −1
 0 1 −1 2 


0 0 1
0
−1 2 1
5
Problem 2. [] The matrix A below has eigenvalues λ1 = −1, λ2 = 2. Find a basis
for R3 consisting of eigenvectors of A.


0 1 1
A = 1 0 1
1 1 0
Problem 3. [] Find the determinant of the matrix:


1 −1 5
5
2
3
1
2
4
5


−1 −3 8
0 −1


1
1
2 −1 2 
−1 1 −3 1 −1
Problem 4. [9] Find all values of b such that the matrix A is NOT invertible:


0 b −1 0
 b 0 0 2

A=
0 1 2 1
5 0 0 5
Problem 5. [5] Find a basis for Span{~v1 , ~v2 , ~v3 , ~v4 , ~v5 }, where
 
 
 
 
1
1
−2
3
2
−1
3
4

 
 
 
~v1 = 
−1 , ~v2 = −2 , ~v3 =  3  , ~v4 =  4  ,
3
−1
2
14
 
2
1

~v5 = 
0 .
5
Problem 6. Let T : R3 → R3 be the linear transformation
T (x1 , x2 , x3 ) = (x1 − 2x2 + 2x3 , 2x1 + x2 + x3 , x1 + x3 ).
a.[3] Find the matrix of the linear transformation T .
b.[3] Find the inverse transformation T −1 expressed the same way as T .
 
x1
x2 
 

Problem 7. [10] Let H be the subspace of R5 consisting of all vectors 
x3  that
x4 
x5
satify the four linear equations:
x1 − 3x2 + 2x4 + 2x5 = 0
−2x1 + 6x2 + x3 + 2x4 − 5x5 = 0
3x1 − 9x2 − 1x3 + 7x5 = 0
−3x1 + 9x2 + 2x3 + 6x4 + −8x5 = 0
Find a basis for H.
Problem 8. [10]
a.[4] Let A and B be some fixed m × n matrices. Prove that
H = {all ~v in Rn such that A~v = B~v }
is a subspace of Rn .
b.[2] Let H be the set R2 with the x-axis removed, but with ~0 put back to H:
x
H={
: y 6= 0} ∪ {~0}.
y
Show that H is not a subspace of R2 . (Find one instance where some axiom
does not hold.)
Problem 9. [10] Mark each statement either True or False. You do not have to
justify your answer.
a.For any m × n matrix A and vector ~b in Rm , the set of solutions of A~x = ~b is a
subspace of Rn .
b.If v is an eigenvector of A then v is also an eigenvector of A2 .
c.For matrices A and B, if AT B is defined, then AB T is also defined.
d.If A and B are 3 × 2 matrices then A · B T always has λ = 0 as an eigenvalue.
e.Let A be a n × n matrix, {~v1 , . . . , ~vk } a basis for N ul(A) and {w
~ 1, . . . , w
~ l} a
basis for Col(A). For some A it is possible that {~v1 , . . . , ~vk , w
~ 1, . . . , w
~ l } forms
a basis for Rn .