A + C =
A-A
=
0
The zero m a t r i x of order m x n is the matrix, denoted 0, w i t h all entries zero.
Properties of a d d i t i o n : For A, B, C, m x n matrices,
^ + 0 = 0+ ^ = ^
'A-A
= O
A + B = B+ A
(O
13
otddl-Vl^
ICJI-C^TVi+M
•
'
(a.cUvVl(Ob 19 C O h l W ^ L ^ - t V U ^ ^
AHB.C)^iA.BnC^^^^^^^^^^
h4
J
)
'
ex. A manufacturing company makes stuffed ammtttls, trains, doll houses, and dolls for local toy
store A. A n order of stuffed animals for toy store A uses 30 square yards of fabric and 20 bags of
cotton filling. A n order of trains uses 10 square yards of aluminum and 1 unit of paint. A n order
of doll houses uses 60 units of wood, 12 boxes of nails, and 7 units of paint. A n order of dolls uses
20 square yards of fabric and 11 bags of cotton filling. Organize this information in a matrix.
do
6
0
A
2(9
0
0
0
II
20
D
CP
ID
0
0
W i t h the impending holiday's, the company decides to increase their order by 15%. Organize
this information i n a matrix.
ISA
-
1.15
SO
20
0
O
GO
0 O O O
o
0
[ 0 O
\0
:f (^0
0
0
o
11.^ I.
0
,23 12-^^ 0
The manufacturing company also makes stuffed animials, trains, doll houses, and dolls for toy
store B with slightly different products. Their information is organized i n the following matrix
f
A P
/V ^
/ 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
D
Compute the total number of units of each component necessary for both toy store^orders.
V
I 34.5 2 3
6
o
Z3>
0
O
12.^
0
0
y
O
11.9 MS O
O
o
^.65
c
o
0
\0
0
20S
3.i'5 0
0
\
2D.<?S
O
O
0
I S
^.0^ ^'
5.2
Matrix multiplication
A row m a t r i x times a column m a t r i x :
( C l
C2
\
= (rici +
r2C2
H
ex.
h
/
(3
-7
p
/
1
^
riCy
+
H
h
r„c„
y
0 \
6
J
6+ ^
M u l t i p l i c a t i o n in general:
21
r2C2
1 \
2
V10
ex.
r„c„) =
<^
hi
If ^ has order_aj }^^Jand B has orde|^>c I then >1B hae-Drdeijn x_^
-4S^--
3^
A
B
5 x z.
ex.
V
ex.
1
2>v
.0
- 1 0
-1
'2
-1
0
/?€^3\
1- 1
0/
y
/-I
'I
\
/ 2 3 \ / 0
-1
2 \
1 1
" 1 0 - 1
V
0
0 ;
V2
-1
B
0
/
57
14 j
A
ex. Solve for x.
ex.
/ 3
5 ^
8
2 ;
I
ex.
2 \
U
5 J
2 ^ ^ 3
/ 0
5y
V 2
I
8
4 2(<?
5 ^
2
y
\
The identity m a t r i x is a n x n matrix, denoted / „ , w i t h Is on the diagonal and Os everywhere
else.
,
1
0
O
4
o
\
O
O
{ O
.0
0
^1
-1
o'
J \ [oj
0
0
•
0
0
o
0
Y
\
I
0
\ U\
-}
z_
-/
0
2
P r o p e r t i e s of multiplication: For A, B, C matrices of appropriate orders
AB=>^BA
A{BC)^iAB)C
in general
( hof
CiJkv\mUtajl'^
(a.^^-oC\odr\\j{\-^
)
A{B + e) = AB + AC M \ 5 V y i W r i o i a
IA = Ai = A^ ( X 19 Hu.
r\AoHfp n o c c t i V / ^ M-ciA-f
ex. A manufacturing company makes stuffed animials, trains, doll houses, and dolls for local toy
store A. A n order of stuffed animals for toy store A uses 30 square yards of fabric and 20 bags of
cotton filling. A n order of trains uses 10 square yards of aluminum and 1 unit of paint. A n order
of doll houses uses 60 units of wood, 12 boxes of nails, and 7 units of paint. A n 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
What is the price per order of stuffed
3S.
a n i m i a l ^ r a i n ^ ^ l l hmise|^and^oUs*^•
0
o
I
ex. Express the following system of equations in terms of matrices.
'Sx
+ 2y + 7z + w =10
8y- z + 3w =8
6x - Ay + I5z - w =12
A = / 5
6
\ -4
^
^
"if - I
15 -
I \
3
f^
\
/
/