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“No one can be told what the matrix is…
they have to see it for themselves…”
- Lawrence Fishburne
It’s a good thing we have Section 7.2a!!!
Definition: Matrix
Let m and n be positive integers. An m x n matrix
(read “m by n matrix”) is a rectangular array of m rows
and n columns of real numbers.
 a11
a
 21


 am1
a12
a22
am 2
a1n 

a2 n 


amn 
We also use the shorthand notation [a ij ] for this matrix.
Definition: Matrix
Each element, or entry, aij , of the matrix uses double subscript
notation. The row subscript is the first subscript i, and the
column subscript is j. The element aij is in the i-th row and
j-th column. In general the order of an m x n matrix is m x n.
If m = n, the matrix is a square matrix. Two matrices are equal
matrices if they have the same order and their corresponding
elements are equal.
Practice Problem!!!
Determine the order of the matrix, and identify the specified
elements.
1 2 3 
2 0 4


Order 2 x 3
a12
a21
= –2
=2
Practice Problem!!!
Determine the order of the matrix, and identify the specified
elements.
1 1
0 4 


 2 1


3 2 
Order 4 x 2
a22
=4
a43
Doesn’t exist!
Matrix Addition and Subtraction
We can add or subtract matrices of the same order by adding
or subtracting their corresponding entries. Matrices of different
orders cannot be added or subtracted!!!
Let
A   aij 
and
B  bij 
be matrices of order m x n.
1. The sum A + B is the m x n matrix
A  B   aij  bij 
2. The difference A – B is the m x n matrix
A  B   aij  bij 
Scalar Multiplication of Matrices
When dealing with matrices, real numbers are scalars. The
product of the real number k and the m x n matrix A   aij 
is the m x n matrix
kA   kaij 
The matrix
kA   kaij 
is a scalar multiple of A.
A Few More Definitions…
Let A = [a ij ] be any m x n matrix. The m x n matrix O = [0]
consisting entirely of zeros is the zero matrix because
A + O = A. In other words, O is the additive identity for the
set of all m x n matrices. The m x n matrix B = [–a ij ] consisting
of the additive inverses of the entries of A is the additive
inverse of A because A + B = O.
Practice Problem!!!
For the given matrices, find (a) A + B, (b) A – B, (c) 3A, and
(d) 2A – 3B.
 1 0 2 
A   4 1 1
 2 0 1 
1 1 2 


A  B  3 1 1 
6 3 0 
2 1 0


B   1 0 2 
 4 3 1
 3 1 2 


A  B   5 1 3
 2 3 2 
Practice Problem!!!
For the given matrices, find (a) A + B, (b) A – B, (c) 3A, and
(d) 2A – 3B.
 1 0 2 
A   4 1 1
 2 0 1 
2 1 0


B   1 0 2 
 4 3 1
 3 0 6 
 8 3 4 


3A  12 3 3 2A  3B   11 2 8


 6 0 3 
 8 9 5 
Practice Problem!!!
Let A = [a ij ] and B = [b ij ] be 2 x 2 matrices with aij = 3i – j and
b ij = i 2 + j 2 – 3 for i = 1, 2, and j = 1, 2.
1. Determine A and B.
2 1
 1 2 
A
B


5 4
 2 5
Practice Problem!!!
Let A = [a ij ] and B = [b ij ] be 2 x 2 matrices with aij = 3i – j and
b ij = i 2 + j 2 – 3 for i = 1, 2, and j = 1, 2.
2. Determine the additive inverse –A of A and verify that
A + (–A) = [0]. What is the order of [0]?
2 1
 2 1
A
A  


5 4
 5 4
0 0
A   A   
  0

0 0
 The order of [0] is 2 x 2.
Practice Problem!!!
Let A = [a ij ] and B = [b ij ] be 2 x 2 matrices with aij = 3i – j and
b ij = i 2 + j 2 – 3 for i = 1, 2, and j = 1, 2.
3. Determine 3A – 2B.
 8 1
3A  2B  

11 2 
Matrix
Multiplication
Definition: Matrix
Multiplication
Let A = [a ij ] be an m x r matrix and B = [b ij ] an r x n matrix.
The product AB = [c ij ] is the m x n matrix where
cij  ai1b1 j  ai 2b2 j 
 air brj
 To multiply two matrices, the columns of the first matrix must
equal the rows of the second matrix. The resulting matrix
has rows and columns determined by the “outside” values.
Ex: Can we multiply a 3 x 2 matrix and a 2 x 4 matrix???
(3 x 2)(2 x 4)
Yes, we can multiply… and the result is a 3 x 4 matrix…
Find the product AB, where possible:
 2 1 3
A

0 1 2 
 2 1  1 0    31
AB  
  0 1  1 0    2 1
 1 4 


B  0 2 
1 0 
 2  4   1 2    3 0 

 0  4   1 2    2  0  
 1 6
AB  

2 2
 Support with a calculator???
Find the product AB, where possible:
 3 4
 2 1 3
B
A


2 1 
0 1 2 
The product AB is not defined!!! Why??
A florist makes three different cut flower arrangements (I, II, and
III). Matrix A shows the number of each type of flower used in
each arrangement.
I II III
Roses  5 8 7 
A = Carnations  6

6
7


Lilies  4 3 3 


The florist can buy his flowers from two different wholesalers
(W1 and W2), but wants to give all his business to one or the
other. The cost of the three flower types from the two wholesalers is shown in matrix B.
W1
W2
Roses 1.50 1.35
B = Carnations 0.95 1.00 


Lilies 1.30 1.35
I
II
Roses  5
A = Carnations  6
III
Roses
8 7
 B = Carnations
6
7


Lilies  4 3 3 
Lilies


W1
W2
1.50 1.35
0.95 1.00 


1.30 1.35
Construct a matrix showing the cost of making each of the three
flower arrangements from flowers supplied by the two different
wholesalers.
We want the columns of A to match up with the rows of B, so we
first switch the rows and columns of A:
Rose Carn Lily
I 5 6 4
II 8

6
3


III 7 7 3 


The new matrix is called the
transpose of A, and is denoted AT
I
II
III
W1
Roses  5
A = Carnations  6
Roses
8 7
 B = Carnations
6
7


Lilies  4 3 3 
Lilies


W2
1.50 1.35
0.95 1.00 


1.30 1.35
Construct a matrix showing the cost of making each of the three
flower arrangements from flowers supplied by the two different
wholesalers.
Now, we find the product AT B:
Rose Carn Lily
I 5 6 4
W1
W2
W1
W2
I 18.40 18.15 
1.35
II 8 6 3  x Carn 0.95 1.00  = II  21.60 20.85






III 7 7 3 
III  21.05 20.50 
Lily 1.30 1.35


Rose 1.50