matrices lecture 1

Consider an arbitrary system of equation in unknown as:
AX = B ………………………………………………………….(1)
? ? ?
?
? ? ?
?
?
+ + + +
+ + + + =
+ + + +
+ + + + =
+ + + + =
+ + + +
m m m mn n
n n
r B n n
a X a X a X a X
am am X m a n n bm
a X a X a X a X
a a a a n n b
ail ai ai ain n b
a X a X a X a X
.....
1 1 2 2 9 3 3 ... 2
.......
21 22 2 23 3 ... 2 2
3 .......
.......
1 1 2 2 3 3
21 1 22 2 23 3 2
1
1 2 2 3 1
1 1 12 2 3 1
? ? ?
? ? ? ?
? ? ? ?
……………………….(2)
The coefficient of the variables and constant terms can be put in the form:
1 1
2
1
1 2
11 12 1 2
1
2 22 2
nx mx
m m mn mxn
n
bm
b
b
n
a a a
a a a n
a a a
? ? ? ? ?
?
?
? ? ? ? ?
?
?
? ? ? ? ?
?
?
? ? ? ? ?
?
?
? ? ?
?
?
? ? ?
?
?
?
?
?
…………………………………..(3)
Let the form
( ) 1
1 2
21 22 2
12 1
n i
u n
A a
am am amn
a a a
a a a
= =
? ? ? ? ? ?
?
?
? ? ? ? ? ?
?
?
…………………………………………...(4)
Is called (mxn) matrix and donated this matrix by:
[aij] i = 1, 2,……m and j = 1, 2,…..n.
We say that is an (mxn) matrix or تكملة
The matrix of order (mxn) it has m rows and n columns.
For example the first row is (a11, a12, a1n)
And the first column is
? ? ? ? ?
?
?
? ? ? ? ?
?
?
1
21
11
12
m a
a
a
a
3
Is (mx1) [m rows and 1 column]
(aij) denote the element of matrix. Lying in the i – th row and j – th column,
and we call this element as the (i,j) - th element of this matrix
Also
1
2
1
n nx
? ? ? ? ?
?
?
? ? ? ? ?
?
?
?
?
?
is (nx1) [n rows and columns]
Sub – Matrix:
Let A be matrix in (4) then the sub-matrix of A is another matrix of A
denoted by deleting rows and (or) column of A.
Let A =
? ? ?
?
?
? ? ?
?
?
7 8 9
4 5 6
1 2 3
Find the sub-matrix of A with order (2×3) any sub-matrix of A denoted by
deleting any row of A ? ??
?
? ??
?
? ??
?
? ??
?
? ??
?
? ??
?
7 8 9
4 5 6
,
7 8 9
1 2 3
,
4 5 6
1 2 3
Definition 1.1:
Tow (mxn) matrices A = [aij] (mxn) and B = [bij] (mxn) are said to be equal
if and only if:
aij = bij for i = 1,2…..m and j = 1,2….n
mx bm
b
b
? ? ? ? ?
?
?
? ? ? ? ?
?
?
2
1
4
Thus two matrices are equal if and only if:
i. They have the same dimension, and
ii. All their corresponding elements are equal for example:
Definition 1.2
If A = [aij] mxn and B = [bij] mxn are mxn matrix their sum is the mxn
matrix A+B =[aij + bij]mxn.
In other words if two matrices have the same dimension, they may be added
by addition corresponding elements. For example if:
A = ?
??
?
? ??
? ?
= ?
??
?
? ??
?
?
?
1 6
5 0
3 4
2 7
and B
Then
A+B = ?
??
?
? ??
?
? + +
+ ? ? +
3 1 4 6
3 5 7 0
= ?
??
?
? ??
?
?
? ?
2 10
3 7
Additions of matrices, like equality of matrices is defined only of matrices
have same dimension.
Theorem 1.1:
Addition of matrices is commutative and associative, that is if A, B and C
are matrices having the same dimension then:
A + B = B + A (commutative)
A + (b + C) = (A + B) + C (associative)
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?
?
? ? ? ?
?
? ? +
= ??
?
??
? ?
2
4
3 20
0(7) 2 1
2
4
3 5 2
2 0 1
5
Definition 1.3
The product of a scalar K and an mxn matrix A = [aij] mxn is the nn,Xn
matrix KA = [kaij] mXn for example:
6 ?
??
?
? ??
?
?
?
= ?
??
?
? ??
?
?
?
= ?
??
?
? ??
?
?
?
30 12 6
6 0 2
6(5) 6(2) 6( 11)
6( 1) 6(0) 6(7)
5 2 11
1 0 7
Application of Matrices
Definition 1.4:
If A = [aij] mxn is mxn matrix and B = [bjk] nxp an nxp matrix, the product
AB is the mxp matrix C = [cik] nxp in which
Cik aij bik
n
j 1 =
? =
Example1: if A =
3 1
2 3 22
21
11
21 22 23
11 12 13
×
? ? ?
?
?4 - Null or Zero Matrix: A matrix each of whose elements is zero is called
null matrix or zero matrix, for example ?
??
?
? ??
?
0 0 0
0 0 0 is a (2×3) null matrix.
5 – Diagonal Matrix: the elements aii are called diagonal of a square matrix
(a11 a22 – ann) constitute its main diagonal A square matrix whose every
element other than diagonal elements is zero is called a diagonal matrix for
Example:
? ? ?
?
?
? ? ?
?
?
0 0 2
0 2 0
1 0 0
or ?
??
?
? ??
?
0 0
0 0
6 – Scalar Matrix: A diagonal matrix, whose diagonal elements are equal, is
called a scalar matrix.
For example
? ? ?
?
?
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?
?
? ? ?
?
?
? ? ?
?
?
? ??
?
? ??
?
0 0 0
0 0 0
0 0 0
,
0 0 1
0 1 0
1 0 0
,
0 5
5 0 are scalar matrix