Section 5.1, 5.2
MATH 166:503
April 7, 2015
Topics from last notes: Annuities, amortization, solving systems of linear equations, representing
a system of linear equations as an augmented matrix, inconsistent systems, systems with infinitely
many solutions, systems with unique solutions
4
SYSTEMS OF LINEAR EQUATIONS AND MODELS
ex. You are trying to guess the number of red gumballs in a jar filled with red, green, and blue
gumballs. you are told that the jar contains 250 gumballs. There are 3 times as many red gumballs
as green gumballs and 5 times as many blue gumballs as green gumballs. Write the augmented
matrix for this problem.
ex. Solve the following system
3x + 2y + 7z + w =10
8y − z + 3w =8
6x − 4y + 15z − w =12
1
5
5.1
ex.
MATRICES
Introduction to matrices


1 2 3 7
M =  −1 0 5 9 
8 4 −3 6
The order of a matrix with m rows and n columns is m x n. In a general matrix A, ai,j is the
entry in the ith row and jth column of A.
In the matrix M above, what is m1,3 ? m2,1 ? m3,1 ? m1,2 ?
ex.


2 x
S= 1 5 
0 y
What is the order of S? What is s1,2 ? s2,2 ? s3,1 ?
2
Special matrix shapes: For some number n,
a matrix of order 1 x n is called a row matrix.
a matrix of order n x 1 is called a column matrix.
a matrix of order n x n is called a square matrix.
ex.


1
 3 

C=
 9 
7
R=
10 −3 0 5 1
S=
1 0
2 3
Two matrices are equal is they have equal order and all the corresponding entries are equal.
Some operations on matrices:
multiplication by a scalar: if c is a number and A = (ai,j ) is a matrix then cA = (cai,j ).
addition of matrices: if A = (ai,j ) and B = (bi,j ) are of the same order, then A + B = (ai,j + bi,j ).
transpose of a matrix: if A = (ai,j ) is an m x n matrix then the transpose, AT = (aj,i ), has order n x m.


8
1 0
A =  −3 1 2 
0 10 5
ex.


−7 0 2
B= 6 1 7 
−5 9 8
3A =
A+B =
3


8
C= 7 
14
BT =
A+C =
2A + B T =
CT =
A − 3B =
A−A=
The zero matrix of order m x n is the matrix, denoted 0, with all entries zero.
Properties of addition: For A, B, C, m x n matrices,
A+0=0+A=A
A−A=0
A+B =B+A
A + (B + C) = (A + B) + C
4
ex. A manufacturing company makes stuffed animials, trains, doll houses, and dolls for local toy
store A. An order of stuffed animals for toy store A uses 30 square yards of fabric and 20 bags of
cotton filling. An order of trains uses 10 square yards of aluminum and 1 unit of paint. An order
of doll houses uses 60 units of wood, 12 boxes of nails, and 7 units of paint. An order of dolls uses
20 square yards of fabric and 11 bags of cotton filling. Organize this information in a matrix.
With the impending holidays, the company decides to increase their order by 15%. Organize
this information in a matrix.
The manufacturing company also makes stuffed animials, trains, doll houses, and dolls for toy
store B with slightly different products. Their information is organized in the following matrix


15 6 0 .5 0 0
 0 0 9 2 0 0 

MB = 
 1 0 0 8 40 3 
17 8 0 1 0 0
Compute the total number of units of each component necessary for both toy store orders.
5
5.2
Matrix multiplication
A row matrix times a column matrix:


c1


 c2 
r1 r2 · · · rn  .  = (r1 c1 + r2 c2 + · · · + rn cn ) = r1 c1 + r2 c2 + · · · + rn cn
 .. 
cn
ex.

3 −7 0
ex.

1
 2 
10

1
3
2
8

0
 6 
−6
Multiplication in general:
If A has order m x n and B has order n x l then AB has order m x l.
6
ex.



0 −1 2
2 3
 −1 0 −1   1 1 
2 −1 0
0 0
ex.
ex. Solve for x.
ex.
ex.



2 3
0 −1 2
 1 1   −1 0 −1 
0 0
2 −1 0
2 −1
7 8
x 2
1 0
=
13 4
57 14
3 5
8 2
0 2
2 5
0 2
2 5
3 5
8 2
The identity matrix is a n x n matrix, denoted In , with 1s on the diagonal and 0s everywhere
else.
7
Properties of multiplication: For A, B, C matrices of appropriate orders
AB 6= BA in general
A(BC) = (AB)C
A(B + C) = AB + AC
IA = AI = A
ex. A manufacturing company makes stuffed animials, trains, doll houses, and dolls for local toy
store A. An order of stuffed animals for toy store A uses 30 square yards of fabric and 20 bags of
cotton filling. An order of trains uses 10 square yards of aluminum and 1 unit of paint. An order
of doll houses uses 60 units of wood, 12 boxes of nails, and 7 units of paint. An order of dolls uses
20 square yards of fabric and 11 bags of cotton filling. The price for one square yard of fabric is
$.50, for a bag of cotton filling is $.25, for a square yard of aluminum is $1, for a unit of paint is
$3, for a unit of wood is $1.50, and for a box of nails is $2. What is the price per order of stuffed
animials, trains, doll houses, and dolls?
ex. Express the following system of equations in terms of matrices.
3x + 2y + 7z + w =10
8y − z + 3w =8
6x − 4y + 15z − w =12
8