Download Day 1

14.1 Matrix Addition and
Scalar Multiplication
OBJ:  To find the sum,
difference, or scalar multiples
of matrices
EX:  An automobile dealer sells four
different models whose fuel economy
is shown in the table below:
City
Mpg
Spts Se- Sta- Van
Car dan tion
Wag
17
22 17
16
High- 23
way
Mpg
30
24
19
This information can
be displayed as a
rectangular array of
numbers enclosed by
brackets,
called a
matrix (plural,
matrices), usually
labeled with a capital
letter.
sp
M =  17
 23
se
22
30
sw
17
24
v
16  c
19  h
Each number is an element (or entry) of the
matrix.
The dimensions are the number of rows and
columns.
Since M has two rows and four columns, M is
a 2 x 4 matrix, denoted by M2x4.
It is a “driving-condition by model” matrix.
If the rows and columns are
interchanged, you get the
transpose of M, denoted by Mt
c
Mt = 
l
l

h

l
l

sp
se
sw
v
Mt 4x2 is a “model by driving-condition”
matrix, with 4 rows and 2 columns.
If the rows and columns are
interchanged, you get the
transpose of M, denoted by M t
c
Mt =  17
l 22
l 17
 16
h
23
30 l
24 l
19 
sp
se
sw
v
Mt 4x2 is a “model by driving-condition”
matrix, with 4 rows and 2 columns.
The Environmental Protection Agency mandated
in 5 years the fuel performance figures must
increase 10%.
This means every element in matrix M must be
multiplied by 1.10, resulting in the matrix
1.1M = 

sp
se
sw
v
18. 7 24.2 18.7 17.6  c
25.3 33 26.5 20.9  h
This is called scalar multiplication, with 1.1 being
called a scalar.
EX:  If A = 3 1 5 , find At, 2A,
and -3A
 4 0 -2 
At=
-3A =
 3 4 
| 1 0 
 5 -2 
2A =
 6 2 10 
8 0 -4 
 -9 -3 -15 
 -12 0
6
Two matrices with the same dimensions
can be added or subtracted, by finding
the sums or differences of the
corresponding elements.
EX: 
A=  3
4
-2
B=  2
4
 0
Find A + B and A – B.
8
0
1
0
-6
7
1
-3
5
9
-5
2
A + B=
5 8 10 
 8 -6 -8 
-2 8 7 
A – B=
 1 8 -8 
 0 6 2 
 -2 -6 3 
EX: 
A = 2 -1
 4 0
0 -8
B = -6 3 5
0 7 -4
Find At + B and A + Bt.
At =
2 4 0 
-1 0 -8 
Bt =
 -6 0 
 3 7 
 5 -4 
At + B =
 -4 7 5 
 -1 7 -12
A + Bt =
-4 -1 
 7 7 
 5 -12
Two matrices are equal if
and only if they have the
same dimensions and all
corresponding elements
(same row, same column)
are equal.
EX:  Find the values
of the variables for
which the given
statement is true.
a b – 2 -3 = 7 2.5
c d 5 -1 -1 0 
a b = 7 2.5 + 2 -3
c d -1 0  5 -1
a b =  9
c d  4
-.5
-1
Solve the matrix equation for X
2 5 1 + 3X =1 -4
3 4
3 -7
 10 2  + 3X = 1 -4
 6 8 
3 -7
3X= 1 -4 – 10 2  =
3 -7
6 8 
_1_ -9 -6 
3 -3 -15
X = -3 -2 
-1 -5 