W2MATH 221 ALL
1. (1 point) Find the value of k for which the matrix
A = 2 4 −5 94 − − −9 7 4 3 −k8 3 5
has rank 2.
k =
Correct Answers:
• -14
2. (1 point) Find the rank and the nullity of the matrix
A = 2 4 −0 0
...
W2MATH 221 ALL
1. (1 point) Find the value of k for which the matrix
A = 2 4 −5 94 − − −9 7 4 3 −k8 3 5
has rank 2.
k =
Correct Answers:
• -14
2. (1 point) Find the rank and the nullity of the matrix
A = 2 4 −0 0 0 1 1 1 2 0 5 1 1 −1 0 −3 −1 3 5:
rank(A) =
nullity(A) =
rank(A)+nullity(A) =
• choose
• the number of columns of A
• the number of rows of A
Correct Answers:
• 3
• 2
• the number of columns of A
3. (1 point) Find all values of x for which rank(A) = 2.
A = 2 4 -1 1 0 5 2 1 x -3 3 0 -2 -8 3 5
x =
Correct Answers:
• -2
4. (1 point)
Suppose that A is a 6 × 5 matrix which has a null space of
dimension 3.
The rank of A is rank(A) =
Correct Answers:
• 2
5. (1 point) Find a non-zero 2×2 matrix such that
−369 7 −28 = 0 0 0 0 .
Correct Answers:
•
−−00::111111 142857 −−00::111111 142857
6. (1 point) Determine if the set of vectors is a basis of R5.
If not, determine the dimension of the subspace spanned by the
vectors.
266664
-1
-1
1
-1
-1
377775
266664
10
-1
-2
1
377775
266664
-2
-1
2
-1
-2
377775
266664
2
-1
-2
-1
2
377775
266664
-1
-2
10
-1
377775
The dimension of the subspace spanned by the vectors is
Correct Answers:
• 3
7. (1 point) Let
B =
2664
3 4 6
−2 1 −8
−3 0 −8
−2 −2 −3
3775
:
(a) Find the reduced row echelon form of the matrix B.
rref(B) =
2664
3775
(b) How many pivot columns does B have?
(c) Do the vectors in the set
8>><>>:
2664
3 −2 −3 −2
3775
;
2664
410 −2
3775
;
2664
6 −8 −8 −3
3775
9>>=>>;
span R4? Be sure you can explain and justify your answer.
• choose
• the vectors span Rˆ4
• the vectors do not span Rˆ4
(d) Are the vectors in the set
8>><>>:
2664
3 −2 −3 −2
3775
;
2664
410 −2
3775
;
2664
6 −8 −8 −3
3775
9>>=>>;
linearly independent? Be sure you can explain and justify your
answer.
• choose
• linearly dependent
• linearly independent
Correct Answers:
2664
1 0 0
0 1 0
0 0 1
0 0 0
3775
• 3
• the vectors do not span Rˆ4
• linearly independent
8. (1 point) Are the following statements true or false?
? 1. If T : R3 ! R9 is a linear transformation, then range
(T) (also known as the image of T) is a subspace of R9.
? 2. The sum of two subspaces of Rn forms another subspace of Rn. The sum of V and W means the set of all
vectors ~v +~w where ~v is an element of V and ~w is an
element of W.
? 3. The intersection of two subspaces of Rn forms another
subspace of Rn.
? 4. If u and v are in a subspace S, then every point on the
line connecting u and v is also in S. [The line is the
set of vectors you can form as tu+(1−t)v for different
values of t]
Correct Answers:
• T
• T
• T
• T
9. (1 point)
If A is an n × n matrix and b 6= 0 in Rn, then consider the set
of solutions to Ax = b.
Select true or false for each statement.
? 1. This set is closed under scalar multiplications
? 2. The set contains the zero vector
? 3. This set is closed under vector addition
? 4. This set is a subspace
Correct Answers:
• FALSE
• FALSE
• FALSE
• FALSE
10. (1 point)
Find the determinant of the matrix
M =
2664
−2 0 0 −1
−1 0 3 0
0 −2 0 2
0 2 2 0
3775
:
det(M) = .
Correct Answers:
• -2*3*2*2--1*-1*-2*2
11. (1 point)
Find k such that the following matrix M is not invertible (singular).
M = 2 4 10−3 2 14 +1 0 k 2 10 −4 3 5
k =
Correct Answers:
• -8
12. (1 point) Given the matrix
A = a+018 a− −018
find all values of a that make det(A) = 0. Give your answer as a
comma-separated list.
Values of a: .
Correct Answers:
• -18, 18
13. (1 point) Evaluate the following 4 × 4 determinant. Use
the properties of determinants to your advantage.
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