Download A • B

4.6 Matrices
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Matrices are rectangular arrangements of data that are used to
represent information in tabular form.
 1 0 4
A

 3 - 6 8
a23
Dimensions: number of rows and columns
A is a 2 x 3 matrix
Elements of a matrix A are denoted by aij.
a23 = 8
2
Data about many kinds of problems can often be represented by
matrix.
e.g Average temperatures in 3 different cities for each month:
23 26 38 47 58 71 78 77 69 55 39 33
A  14 21 33 38 44 57 61 59 49 38 25 21
3 cities
35 46 54 67 78 86 91 94 89 75 62 51
12 months Jan - Dec
Average temp. in the 3rd
city in July, a37, is 91.
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Matrix of coefficients
Solutions to many problems can be obtained by solving
systems of linear equations.
For example, the constraints of a problem are represented
by the system of linear equations
x + y = 70
24x + 14y = 1180
1 1
The matrix A  

24
14


is the matrix of coefficient for this system of linear
equations.
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In a matrix, the arrangement of the entries is significant.
Therefore, for two matrices to be equal they must have
the same dimensions and the same entries in each
location.
Example: Let
 x 4
X  1 y 
 z 0 
3 4 
Y  1 6 
2 w
If X = Y, then x = 3, y = 6, z = 2, and w = 0.
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Square Matrix is a matrix in which the number of
rows equals the number of columns.
•Main Diagonal: in a n x n square matrix, the elements
a11, a22, a33, …, ann form the main
diagonal of the matrix.
•Symmetric matrix:If the corresponding elements
match when we think of folding the
matrix along the main diagonal, then
the matrix is symmetric about the main
diagonal.
In a symmetric matrix, aij = aji.
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Example: The square matrix
1 5 7 
A  5 0 2
7 2 6
Main Diagonal
is symmetric.
Note that
a21 = a12 = 5
a31 = a13 = 7
a32 = a23 = 2
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Matrix Operations
1. Scalar multiplication:
Multiply each entry of a matrix by a fixed single
number called scalar.
ex: The result of multiplying matrix
1 4 5
A

6
3
2


by the scalar r = 3 is
 3 12 15
A

18 - 9 6 
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2. Addition:
Adding the corresponding elements of 2 matrices
that have the same dimensions.
ex: For
 1 3 6
A   2 0 4
 4 5 1
0 - 2 8 
B  1 5 2
2 3 3
the matrix A+B is
 1 1 14
A  B   3 5 6 
 2 8 4 
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3. Subtraction:
defined by A – B = A + (-1)B.
In a zero matrix, all entries are 0.
An n  m zero matrix is denoted by 0.
If A and B are n x m matrices and r and s are
scalars, the following matrix equations are true:
0+A=A
A+ B =B +A
(A + B) + C = A + (B + C)
r(A + B) = rA + rB
(r + s)A = rA + sA
r(sA) = (rs)A
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4. Multiplication of matrices:
A:
B:
n  m matrix
m  p matrix
The result C is an n  p matrix.
A  B = C, where cij 
m
 aik bkj
k 1
An entry in row i, column j of A  B is obtained
by multiplying elements in row i of A by the
corresponding elements in column j of B and adding
the results.
To compute A times B, the number of columns in A
must equal the number of rows in B.
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Example: Let
 2 4 3
A

4

1
2


2  3 matrix
5 3
 2 4 3 
 36

4  1 2 2 2   __

 6 5  


__ 
__ 
5 3
 2 4 3 
 36

4  1 2 2 2   __

 6 5  


29 
__ 
 5 3
 2 4 3 
36 29

4  1 2 2 2  30 20

 6 5  



5 3
B  2 2
6 5
3  2 matrix
2(5) + 4(2) + 3(6)
= 10 + 8 + 18
= 36
Note that A is a 2  3 matrix
and B is a 3  2 matrix. The
product A • B is a 2  2 matrix.
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Example: Compute A  B and B  A for
3 6
1 4 
B
A


3
4


6  2
1(6)  4(4)  15 22
 1(3)  4(3)
A B  



6
(
3
)

(

2
)(
3
)
6
(
6
)

(

2
)(
4
)
12
28

 

3(1)  6(6) 3(4)  6(2) 39 0
B A 



3(1)  4(6) 3(4)  4(2) 27 4
Note: A  B  B  A .
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Where A, B and C are matrices of appropriate
dimensions and r and s are scalars, the following
matrix equations are true:
(The notation A(B  C) is shorthand for A  (B  C) )
A(B  C) = (A  B)C
A(B + C) = A  B + A  C
(A + B)C = A  C + B  C
rA  sB = (rs)(A  B)
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Identity matrix
The n  n matrix with 1s along the main diagonal and 0s
elsewhere is called the identity matrix, denoted by I. If
we multiply I times any n  n matrix A, we get A as the
result. The equation
I•A=A•I=A
holds.
Let
1 0
I 

0
1


 a11
A
a 21
a12 
a 22 
1(a11 )  0(a21 ) 1(a12 )  0(a22 )  a11 a12 
IA

A


0(a11 )  1(a21 ) 0(a12 )  1(a22 ) a21 a22 
Similarly, A • I=A.
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An n  n matrix A is invertible if there exists an n  n
matrix B such that
A• B = B •A= I
In this case B is called the inverse of A, denoted by A-1.
Let
 1 2  3
A   2
1
0 
 4  2 5 
 5 4  3
B   10  7 6 
 8  6 5 
Then, following the rules of matrix multiplication, it can
be shown that A • B = B • A = I, so B = A-1.
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Boolean Matrices
Matrices with only 0s and 1s as entries are called
Boolean matrices.
Boolean multiplication: x  y = min(x,y)
Boolean addition:
x  y = max(x,y)
Boolean matrix multiplication A  B is defined by
m
Cij =  (aik  bkj)
k=1
A  B: corresponding elements are combined using
Boolean multiplication.
A  B: corresponding elements are combined using
Boolean addition.
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Let A and B be Boolean matrices,
1 1 0 
A  0 1 0
0 0 1
Then
1 0 0 
B  1 1 1
0 0 1
1 1 0
A  B  1 1 1
0 0 1
1 0 0
A  B  0 1 0
0 0 1
And the Boolean product A B is
1 1 1
A  B  1 1 1
0 0 1
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