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Matrix
Definition: An array of numbers in m rows and n colums is
called an mxn matrix
 a11 a12  a1n 
a

a

a
22
2n 
A   21
 


 


am1 am 2  amn 
A square matrix of order n, is an (nxn) matrix
Matrix Equality
Two matrices A = [aij] and B = [bij] are equal if they have the
same order and aij = bij for every i and j.
For example,
0.5

1
 4
.

9  1
3  since both matrices
 2
7  0.25 7 
 

are of order 2x2 and all corresponding entries are equal.
Matrix Addition
Definition: The sum of two matrices is the matrix where
each element is the sum of the corresponding elements
of the two matrices.
NOTE: Matrix addition is only defined when the matrices
have the same dimension.
Examples:
1 2 3 7 2 1  1  2 2  2 3  1  8 4 4 
 4 5 6  1 3 4    4  1 5  3 6  4   5 8 10 

 
 
 

Matrix Substraction
Definition: The substraction of two matrices is the matrix
where each element is the difference of the corresponding
elements of the two matrices.
NOTE: Matrix substraction is only defined when the matrices
have the same dimension.
Example:
2 4 1 0  2  1 4  0 1 4 
5 1  3 7  5  3 1  7   2  6

 
 
 

Matrix Addition
Theorem 1:
Let A, B, and C be matrices of the same size, then
A B  B  A
Commutative law of addition
( A  B )  C  A  ( B  C ) Associative law of addition
Scalar Multiplication
•
Definition: Let a be a number and A=(aij)mxn be any
matrix. Then
aA  Aa  (aaij ) mxn
•
This means that aA is the matrix by multiplying each
entry of A by the same number a
1 2 5 3 6 15 
3



0 6 7 0 18 21
Negative of a Matrix
•
Definition: Let A=(aij) be a mxn matrix. The negative of
the matrix A is the mxn matrix B=(bij) such that bij=-aij for
all i,j.
The negative of A is written as –A.
For example,
4 0
0
1
 1  4
If A   3  5 9 then  A    3
5  9
7
 7  2  8
2 8
Zero Matrix
•
Definition: The mxn matrix in which every entry is 0 is
called the mxn zero matrix and is denoted by 0mxn or
simply 0.
•
We note that A+(-A)=0, the zero matrix.
•
We usualy write A+(-B) as A-B
Multiplication of Matrices
•The product of two matrices A and B is defined if A is mxn
and B is nxp
•We first give the rule for multplication in special case
when A is 1xn and B is nx1
b 
1
Let A  a1 a2  an 
 
b2 
and B   
 
b 
 n
Then their product AB is defined to be a 1x1 matrix,
AB  a1b1  a2b2    anbn
Multiplication of Matrices
Now we give the rule for multiplication of matrices in general.
Definition: Let A=(aij)mxn and B=(bij)nxq. Then
 b1 j 
 
b2 j 

Where cij is the product of ai1 ai 2  ain  and  

 
 bnj 
That is cij  ai1b1 j  ai 2b2 j    ainnj
AB =(cij)mxp
Multiplication of Matrices


1
2 3
4
5 6



10
20
30
12
22
32

1*10+2*20+3*30
1*12+2*22+3*32
4*10+5*20+6*30
4*12+4*22+6*32
# of Columns of A must = # of Rows of B

Diagonal of a Matrix
Definition: If A is an nxn matrix, then the line joining (1,1)
entry, (2,2) entry,…,(n,n) entry is called the diagonal (or
the main diagonal) of the matrix.
Definition: Any nxn matrix D is called diagonal matrix if
each entry not on the diagonal is 0.
For example,
 d1 0  0 
0 d  0 

D 2
 



0 0  dn 
Identity Matrix
Definition: An nxn matrix A=(aij), where aij=1 whenever i=j,
and aij=0 whenever i  j is called the identity matrix of
order n. And it is denoted by I n
Square matrix with ones on the diagonal and zeros
elsewhere.
1 0
I2   
0 1
1 0 0
I 3  0 1 0
0 0 1
1 0 0 0
0 1 0 0 

I4  
0 0 1 0 


0
0
0
1


It can be easily proved that if A is an mxn matrx, then
I m A  A  AI n
Properties of Matrix Multiplication
1) A( BC )  ( AB)C
2) A( B  C )  AB  AC
3) ( B  C ) A  BA  CA
4) r ( AB)  (rA) B  A(rB ) for any scalar r
5) I m A  A  AI n for m  n matrix A
In general the followings are NOT true.
AB  BA
If AB  AC then B  C
If AB  0 then A  0 or B  0
Transpose
Definition: Given an mxn matrix A, the transpose of A
is the nxm matrix, denoted by AT, whose columns are
formed from the corresponding rows of A.
A  (aij )  A  (a ji )
T
 a11 a12  a1n 
 a11
a

a
a

a
22
2n 
If A   21
Then AT   12
 
 


 



a
a

a
m2
mn 
 m1
a1n
a21  am1 
a22  am 2 


 

a2 n  amn 
In other words, the ith row of AT is the ith column of A for all i.
 3 1 0
Example: Let A : 


1
2
4


What is AT ?
Rules related to transpose:
1) ( AT )T  A
2) ( A  B)T  AT  BT
3) (rA)T  rAT for any scalar r
4) ( AB)T  BT AT
Matrix Powers
• Recall that matrix multiplication is associative, i.e. if
A, B and C have the proper dimensions, then
A ( BC )  ( AB ) C, so the parentheses are
unnecessary and the product can be written as ABC.
• If A is an n x n matrix and p is a positive integer, can
define
A p  AA
A
p factors
• Again, if A is an n x n matrix, adopt the convention
A0  In
Triangular Matrices
• An n x n matrix A  [ aij ] is called upper triangular if
aij  0 for i > j
• An n x n matrix A  [ aij ] is called lower triangular if aij
 0 for i < j
• Note:
– A diagonal matrix is both upper and lower
triangular
– The n x n zero matrix is both upper and lower
triangular
Symmetry
• Defn - A matrix A is called symmetric if AT A
• Defn - A matrix A is called skew-symmetric if AT A
• Comment - If A is skew-symmetric, then the diagonal
elements of A are zero
• Comment - Any square matrix A can be written as
the sum of a symmetric matrix and a skew-symmetric
matrix



A  1 A  AT  1 A  AT
2
2
symmetric

skew-symmetric
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