Application on matrices and system of
linear equations
practice problems
Dr. Nikunja Bihari Barik
Department of Mathematics
1. The accompanying figure shows the flow of traffic near Vijayawada
Market complex during the rush hours on Sunday. The arrows indi-
cate the direction of traffic flow on each one-way road, and the average
number of vehicles per hour entering and leaving each intersection ap-
pears beside each road. 5th Avenue and 6th Avenue can each handle
up to 2000 vehicles per hour without causing congestion, whereas the
maximum capacity of both 4th Street and 5th Street is 1000 vehicles
per hour. The flow of traffic is controlled by traffic lights installed at
each of the four intersections.
(a) Write a general expression involving the rates of flow x1 , x2 , x3 , x4
and suggest two possible flow patterns that will ensure no traffic con-
gestion.
(b) Suppose the part of 4th Street between 5th Avenue and 6th Avenue
is to be resurfaced and that traffic flow between the two junctions must
therefore be reduced to at most 300 vehicles per hour. Find two possible
flow patterns that will result in a smooth flow of traffic.
2. In the downtown section of Bhubaneswar two sets of one-way streets
intersect. The average hourly volume of traffic entering and leaving this
section during rush hour is given in diagram. Determine the amount
of traffic between each of the four intersection.
3. Balance the chemical equation and derive the balanced equation by
using system of linear equations.
P Cl5 + H2 O → H3 P O4 + HCl.
4. A dietitian wishes to plan a meal around three foods. The meal is to
include 8800 units of vitamin A, 3380 units of vitamin C, and 1020 units
of calcium in each ounce of the foods is summarized in the following
table:
Determine the amount of each food the dietitian should include in the
meal in order to meet the vitamin and calcium requirements.
5. Ace Novelty received an order from Magic World Amusement Park
for 900 ”Giant Pandas,” 1200 ”Saint Bernards,” and 2000 ”Big Birds.”
2
Ace’s management decided that 500 Giant Pandas, 800 Saint Bernards,
and 1300 Big Birds could be manufactured in their Los Angeles plant,
and the balance of the order could be filled by their Seattle plant. Each
Panda requires 1.5 square yards of plush, 30 cubic feet of stuffing, and
5 pieces of trim; each Saint Bernard requires 2 square yards of plush,
35 cubic feet of stuffing, and 8 pieces of trim; ach Big Bird requires 2.5
square yards of plush, 25 cubic feet of stuffing, and 15 pieces of trim.
The plush costs $4.50 per square yard, the stuffing costs 10 cents per
cubic foot, and the trim costs 25 cents per unit.
1 −1 0 0 1
6. Sketch the nodal incidence matrices −1 1 −1 1 0 .
0 0 1 −1 0
7. Find the nodal incidence matrix of the following electrical network.
8. Supercomp Ltd produces two computer models PC1086 and PC1186.
The matrix A shows the cost per computer (in thousands of dollars)
and B the production figures for the year 2010 (in multiples of 10,000
units). Find a matrix C that shows the shareholders the cost per quar-
ter (in millions of dollars) for raw material, labor, and miscellaneous.
3
9. Find the current of the following electrical network.
10. Show that v1 , v2 , v3 are independent but v1 , v2 , v3 , v4 are dependent:
1 1 1 2
v1 = 0 ,v2 = 1 v1 = 1 ,v1 = 3
0 0 1 4
11. Find the largest possible number of independent vectors among
1 1 1 0 0
−1 0 0 1 1
0 ,v2 = −1
v1 = ,v3 =
0 ,v4 = −1 ,v5 = 0
,
0 0 −1 0 −1
4
0
0
v6 =
1
−1
12. Decide the dependence or independence of
(a) the vectors (1, 3, 2), (2, 1, 3) and (3, 2, 1).
(a) the vectors (1, −3, 2), (2, 1, −3) and (−3, 2, 1).
