MAT 202, Homework #1, Summer 2016 Name _______________________________________________
Instructions: Write your work up neatly and attach to this page. Use exact values unless specifically
asked to round. Show all work.
1. Solve the system of equations using elementary row operations on the augmented matrix for
the system. State your solution in vector form, or state that the solution is inconsistent (no
solution). If the solution is consistent (has a solution), state whether it is dependent (infinitely
many solutions) or independent (one solution). If the system is dependent, write the solution in
parametric form. (For four and five variables systems, you may use technology; two and three
variable systems should be solved by hand.)
a.
3x1  6 x2  3

5 x1  7 x2  10
 6 x3  8
2 x1

b. 
x2  2 x3  3
3x  6 x  2 x  4
2
3
 1
 4 x4  10
 2 x1

3 x2  3 x3
0

c. 
x3  4 x4  1

 3 x1  2 x2  3 x3  x4  5
2 x1  5 x2  8 x3  0
d. 
3x1  4 x2  2 x3  0
 x1  2 x2  3 x3  0

x2  2 x3  0
e. 
2 x  4 x  9 x  0
2
3
 1
f.
 4 x4  x5  0
 2 x1

3 x2  3 x3
 x5  0


x3  4 x4  6 x5  0

 3 x1  2 x2  3 x3  x4  2 x5  0
2
 
2. Construct a system of equations with three variables such that the vector 1 𝑡 is a solution of
 
 1 
the system.
1
1
1
3. Solve the systems below by substituting 𝑎 = 𝑥 , 𝑏 = 𝑦 , 𝑐 = 𝑧. Be sure to state your final solution
in the original 𝑥, 𝑦, 𝑧 variables.
a.
b.
12
12
− 𝑦 =7
𝑥
{3 4
+𝑦 =0
𝑥
2
1
2
+ − =5
𝑥
𝑦
𝑧
3
4
− 𝑦 = −1
𝑥
2
1
3
{𝑥 + 𝑦 + 𝑧 = 0
4. Determine the value of any variables (g, h, k) in the system or augmented matrix to make the
system have the indicated type of solution.
4𝑥 + 𝑘𝑦 = 6
a. {
, infinitely many solutions
𝑘𝑥 + 𝑦 = −3
𝑘𝑥 + 2𝑘𝑦 + 3𝑘𝑧 = 4𝑘
b. { 𝑥 + 𝑦 + 𝑧 = 0 , exactly one solution
2𝑥 − 𝑦 + 𝑧 = 1
𝑥 + 2𝑦 + 𝑘𝑧 = 6
c. {
, no solution
3𝑥 + 6𝑦 + 8𝑧 = 4
1
2

 4
e. 
2
d.
f.
5
, consistent
8 6 
12 h 
, consistent
6 3
h
 1 4 7 g 
 0 3 5 h  , consistent


 2 5 9 k 
5. For each statement below determine if it is true or false. If the statement is false, briefly explain
why it is false and give the true statement.
a. Every elementary row operation is reversible.
b. A 5x6 matrix has six rows.
c. The solution set of a linear system involving variables x1 ,..., xn is a list of numbers ( s1 ,..., sn )
that makes each equation in the system a true statement when the values s1 ,..., sn are
substituted for x1 ,..., xn respectively.
d. Two fundamental questions about a linear system involve existence and uniqueness.
e. Two equivalent linear systems can have different solution sets.
f.
The row reduction algorithm applies only to augmented matrices for a linear system.
g. Finding a parametric description of the solution set of a linear system is the same as solving
the system.
h. If one row in an echelon form of an augmented matrix is 0 0 0 5 0 , then the
associated linear system is inconsistent.
i.
If every column of an augmented matrix has a pivot, then the system is consistent.
j.
The pivot positions in a matrix depend on whether row interchanges take place.
k. Whenever a system has free variables, the solution set contains many solutions.
l.
A homogeneous equation is always consistent.
6. Row reduce the following matrices to echelon and reduced echelon form (report both versions).
Explain why echelon form is not unique but reduced echelon form is.
1 2 4 5 
a.  2 4 5 4 


 4 5 4 2 
1
0
b. 
0

0
0 9 0
1
3
0
0
0
0
4
0 1
1 7 

0 1
7. Give 5 examples of a 4x5 matrix in echelon form. Use only 0, 1, * to fill your matrix, with 1
marking the pivot points. For each matrix, indicate whether the solution is consistent or
inconsistent; and if consistent whether it is independent or dependent. For dependent systems,
also list the number of free variables.
8. Find an appropriate interpolating polynomial 𝑝(𝑥) = 𝑎0 + 𝑎1 𝑥 + 𝑎2 𝑥 2 + 𝑎3 𝑥 3 + ⋯ 𝑎𝑛 𝑥 𝑛 or
other indicated equation for the given data. Recall that you are creating a system of equations
that can be solved for the unknown coefficients. Use substitutions as appropriate.
a. (1,7), (2,17), (3,31), (4,65), cubic
b. (−2,28), (−1,0), (0, −6), (1, −8), (2,0), quartic
c. (1,3), (−2,6), (4,2), circle (𝑥 − ℎ)2 + (𝑦 − 𝑘)2 = 𝑟 2
d. (−5,1), (−3,2), (−1,1), (−3,0), ellipse,
(𝑥−ℎ)2
𝑎2
+
𝑖
9. Construct a 3x3 matrix A having 𝑎𝑖𝑗 = 𝑗.
𝑖 + 𝑗, 𝑖 > 𝑗
10. Construct a 3x3 matrix D having 𝑑𝑖𝑗 = { 0, 𝑖 = 𝑗 .
𝑖 − 𝑗, 𝑖 < 𝑗
(𝑦−𝑘)2
𝑏2
=1
11. Write a matrix equation that determines the loop currents. You may solve the systems by
calculator if you wish.
a.
b.
c.
12. Consider the following matrices.
3 4
1
 0 3 5 
 3 1
 9 3
 2 2 


A 
,B  
,C  
, D  2 1 0 , E   1 4 0 







 1 4 
 1 0
 4 4 
 3 4 1 
 1 2 7 
 6 7 
 1 3 2 0
4
F
, G  11 5 , H  1 0 2  , J   



 2 4 1 5
 1
 2 3 
a. Add
i.
ii.
A+B
2B-3C
b. Multiply
i. 4A
ii. -5H
c. Multiply. If the multiplication is not possible, explain why not.
i. AB
ii. BA
iii. DE
iv. BF
v. GC
vi. DG
vii. EH
viii. BJ
13. For each of the traffic networks below, set up a system of equations to solve it. Solve the
system with technology. Is there a single solution?
a.
b.
c.