Kaisa Catt
MATH 204 WINTER 2023
Assignment Practice Problems NOT MANDATORY due 04/17/2023 at 11:59pm EDT
1. (1 point) A square matrix is called a permutation matrix
if it contains the entry 1 exactly once in each row and in each
column, with all other entries being 0. All permutation matrices
are invertible. Find the inverse of the permutation matrix


0 0 1 0
 0 0 0 1 

A=
 1 0 0 0 .
0 1 0 0



A−1 = 




4. (1 point) Solve the system

+2x3 +4x4 = 9

 x1

x2 −3x3 −2x4 =−21
2x1 −3x2 +10x3 +14x4 = 66



−x2 +3x3 +8x4 = 27
x1 =
x2 =
x3 =
x4 =
Correct Answers:
•
•
•
•
-5
-4
5
1
Correct Answers:
•

0
 0

 1
0
0
0
0
1
1
0
0
0

0
1 

0 
0

2. (1 point)



Select all statements below which are true for all invertible n × n
matrices A and B
5. (1 point) Solve the system

4x1 −5x2 +4x3 +4x4 = 0



−x1 +x2 +2x3 +2x4 = 3
3x1 −4x2 +6x3 +6x4 = 3



3x1 −3x2 −6x3 −6x4 =−9
 





x1






x2 
=
 +s 
 +t 
.





x3  
x4
Correct Answers:
•
•
•
•
•
•
(A + A−1 )3
= A3 + A−3
• [[-15],[-12],[0],[0]],


14
 12 


 1 
0
,


14
 12 


 0 
1
A.
B. (A + B)2 = A2 + B2 + 2AB
C. (In − A)(In + A) = In − A2
D. A + In is invertible
E. 8A is invertible
F. ABA−1 = B
Correct Answers:
• CE
3. (1 point)
Determine the value of k for which the system
6. (1 point) Find the determinant of the matrix


−3 4 5 −5
 −5 −3 5 0 
.
C=
 3
5 0 5 
1 −5 0 4

 x +y+4z= −3
x +2y−3z= −3

5x+11y+kz=−14
has no solutions.
k=
det(C) =
.
Correct Answers:
Correct Answers:
• -705
• -22
1
11. (1 point)
Which of the following sets are subspaces of R3 ?
7. (1 point)
Find k such that the following matrix M is singular.


3
1
4
−6
−5 −8 
M=
−29 + k −8 −20
•
•
•
•
•
•
k=
Correct Answers:
• 14
A. {(x, y, z) | x, y, z > 0}
B. {(x, y, z) | 2x − 9y = 0, 7x + 6z = 0}
C. {(x, y, z) | − 2x + 9y − 7z = 0}
D. {(x, y, z) | x + y + z = −6}
E. {(−2, y, z) | y, z arbitrary numbers }
F. {(6x − 8y, −3x − 4y, 5x + 4y) | x, y arbitrary numbers
}
Correct Answers:
8. (1 point)
Find the determinant of the matrix

−1 0
0 −3
 −1 0
1
0

0
−1
0
0
M=

 0
0
0 −3
0 −3 −1 0
det(M) =
• BCF
0
0
3
−1
0
12. (1 point)
Which of the following sets of vectors are linearly independent? (Check the boxes for linearly independent sets.)



.


•
.
Correct Answers:
•
• -1*1*3*-3*-3+-3*-1*-1*-1*-1
•
9. (1 point)
Find the determinant of the n × n matrix A with 4’s on the
diagonal, 1’s above the diagonal, and 0’s below the diagonal.
•
•
det(A) =
.
•
Correct Answers:
  
 

−6
−7 
 3
A.  8  ,  −5  ,  0 


2
−9
4

 
 

3
−8 
 −5
B.  2  ,  5  ,  −3 


−4
−9
5
−4
5
−2
C.
,
,
3
−9
8
−2
2
D.
,
8
−8
0
−5
E.
,
0
9
6
F.
−7
• 4ˆn
Correct Answers:
• AF
10. (1 point)
Which of the following subsets of R3×3 are subspaces of
3×3
R ?
13. (1 point) Find a linearly independent set of vectors that
spans the same subspace of R4 as that spanned by the vectors

 
 
 

2
−2
−6
−2
 −1   −1   1   −3 

 
 
 

 −2  ,  0  ,  4  ,  −2  .
−2
−2
2
−6
• A. The 3 × 3 matrices with trace 0 (the trace of a matrix
is the sum of its diagonal entries)
• B. The 3 × 3 matrices in reduced row-echelon form
• C. The non-invertible 3 × 3 matrices
• D. The 3 × 3 matrices with all zeros in the second row
• E. The 3 × 3 matrices whose entries are all integers
 
8
• F. The 3 × 3 matrices A such that the vector  2  is
1
in the kernel of A
A linearly independent spanning set for the subspace is:

 




 
, 
 



(
Correct Answers:
• ADF
)
Correct Answers:
• [[2],[-1],[-2],[-2]],
2
.

−2
 −1 


 0 
−2

18. (1 point) Let
A=
14. (1 point)
The set
B=
is a basis for
R2 .
5
1
5
,
5
0.25
−5
.
Find an invertible matrix S and a diagonal matrix D such that
S−1 AS = D.
Find the coordinates of the vector ~x =
15
7
relative to the basis B.
[~x]B =
S=
D=
Correct Answers:
•
2
1
Correct Answers:
•
15. (1 point) Find the eigenvalues of the matrix


65 −20 −140
0
10  .
C =  −5
30 −10 −65
•
.
The eigenvalues are
(Enter your answers as a comma separated list. The list you
enter should have repeated items if there are eigenvalues with
multiplicity greater than one.)
1
−8
1
4
−6
0
0
−3
19. (1 point) Find a unit vector with positive first coordinate
that is orthogonal to the plane through the points P = (-5, -1, 3),
Q = (-4, 0, 4), and R = (-4, 0, 9).
(
,
,
)
Correct Answers:
• -5, 0, 5
Correct Answers:
• 0.707106781186547
• -0.707106781186547
• 0
16. (1 point) Find the eigenvalues of the matrix


4
0
0
C =  45 −5 0  .
27 0 −5
20. (1 point) Consider the planes 2x + 3y + 2z = 1 and
2x + 2z = 0.
(A) Find the unique point P on the y-axis which is on both
planes. ( ,
,
)
(B) Find a unit vector u with positive first coordinate that is
parallel to both planes.
i+
j+
k
(C) Use parts (A) and (B) to find a vector equation for the
line of intersection of the two planes,r(t) =
i+
j+
k
The eigenvalues are
.
(Enter your answers as a comma separated list. The list you
enter should have repeated items if there are eigenvalues with
multiplicity greater than one.)
Correct Answers:
• -5, -5, 4
17. (1 point) Let
M=
3
2
−1
6
.
Correct Answers:
Find formulas for the entries of M n , where n is a positive integer.
Mn =
−4
8
Correct Answers:
•
2 ∗ 4n − 5n
(−2) ∗ 4n − (−2) ∗ 5n
•
•
•
•
•
•
•
•
•
(−1) ∗ 5n − (−1) ∗ 4n
2 ∗ 5n − 4n
3
0
0.333333333333333
0
0.707106781186547
0
-0.707106781186547
t*2/sqrt( 2**2 + 2**2 )
1/3
t*(-2)/sqrt( 2**2 + 2**2 )
26. (1 point)
Determine whether the product Ax is defined or undefined.
21. (1 point) Consider the two lines
L1 : x = −2t, y = 1 + 2t, z = 3t
and
L2 : x = −9 + 5s, y = 5s, z = 1 + 5s
Find the point of intersection of the two lines.
,
P=(
,




8 5
-5

6 4 
,x = -9 
9 -4 
-2
8 4


2
-9 1 -9
 -8
-2 -9 3  ,x =
 -8
7 2 8

 8
4
 -6 
5 -5 9

? 3. A =
,x =
 1 
-9 -6 2


 -8
4 -6
-7
3  ,x = -2 
? 4. A = 7
-7 -10
-5

-3
 7
? 1. A =
 -6
-9

-7
? 2. A = -4
-1
)
Correct Answers:
• -4
• 5
• 6
22. (1 point) Find an equation of the plane through the point
(-4, -4, 1) and perpendicular to the vector (-1, 0, 0). Do this
problem in the standard way or WebWork may not recognize a
correct answer.
x+
y+
z=
Correct Answers:
•
•
•
•
-1
0
0
4
? 5. A = -4
23. (1 point) Find the angle in radians between the planes
2x + z = 1 and −1y + z = 1.
0
-2
0
-10


,x =






-3
-1
1
6
-7






Correct Answers:
• Defined
• Defined
• Undefined
• Undefined
• Defined
Correct Answers:
• 1.24904577239825
24. (1 point) Find the distance from the point (3, -5, -3) to
the plane 5x − 1y − 2z = 5.
Correct Answers:
27. (1 point) Consider the two lines
L1 : x = −2t, y = 1 + 2t, z = 3t
and
L2 : x = −8 + 4s, y = 2 + 3s, z = 2 + 4s
Find the point of intersection of the two lines.
• 3.83405790253616
25. (1 point)
If a = i + 6j + k and b = i + 9j + k, find a unit vector with
positive first coordinate orthogonal to both a and b.
i+
j+
k
P=(
Correct Answers:
,
,
)
Correct Answers:
• -4
• 5
• 6
• 0.707106781186547
• 0
• -0.707106781186547
Generated by c WeBWorK, http://webwork.maa.org, Mathematical Association of America
4