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Transcript of a a adeltauniv.edu.eg/new/engineering/wp-content/uploads/Presentation-… · Bz A(BC) = (AB)C A(B +...
Definition (Matrix): A matrix is a set of real or complex numbers (or
elements) arranged in some rows and columns which
form a rectangular array.
The numbers in the array are called entries or elements
of the matrix. If a matrix has m rows and n columns,
we say that its dimension (order) is m by n m n
11 12 1
21 22 2
1 2
n
n
m m mn
a a a
a a a
a a a
A
Matrices
1n
1 2 na a a
is called a column vector
1 nis called a row vector
na
a
a
2
1
Operations on matrices
1) Equality of matrices:
2) Matrix addition and subtraction
11 12 1
21 22 2
1 2
p
pij m p
m m mp
ka ka ka
ka ka kak ka
ka ka ka
A
3) Scalar multiple of a matrix:
Example
If
9 1 3 4 7 8
2 4 2 , 9 3 5 ,
7 1 5 1 1 2
A B Find 3A-5B.
Solution
27 3 9
3 6 12 6 ,
21 3 15
C
A
20 35 40
5 45 15 25
5 5 10
D
B
7 32 31
3 5 39 3 31
26 8 25
C D
A B
A B B A
Properties of matrix addition and scalar
Multiplication:
A (B C) (A B) C
1 2 1 2( ) ( ) A A
1 2 1 2( ) A A A
1 1 1( ) A B A B
1 A A
11 12 1 11 12 1
21 22 2 21 22 2
1 21 2
p n
p n
p p pnm m mp
a a a b b b
a a a b b b
b b ba a a
AB
Matrix Multiplication
11 11 1 1 11 1 1
21 11 2 1 21 1 2
1 11 1 1 1
... ...
... ...
... ...
p p n p pn
p p n p pn
m mp p m n mp pn
a b a b a b a b
a b a b a b a b
a b a b a b a b
Example
Find the product of the following matrices:
4 7 9 2
3 5 6 8
A Β
Solution
4 9 7 6 4 ( 2) 7 8
3 9 5 6 3 ( 2) 5 8
AB78 48
=57 34
Example
Find the product of the following matrices:
5 84 3
1 02 0
2 7
A B
Solution
5 ( 4) 8 2 5 ( 3) 0 8
1 ( 4) 0 2 1 ( 3) 0 0
2 ( 4) 7 2 2 ( 3) 0 7
AB
4 15
4 3
6 6
m p p n m n A B C
Remarks:
BA AB
A(BC) = (AB)C
A(B + C) = A B + A C
(B + C)A = BA + CA
11 21 1
12 22 2
1 2
m
mT
n n mn
a a a
a a a
a a a
A
Transpose of a matrix:
Example
51 0 2 2 1 1
, , 03 2 1 2 0 3
3
A B C
Solution
1 3
0 2 ,
2 1
T
A 1 1 1
1 2 2
TT
A B
1 1
1 2 ,
1 2
5 0 3 .T C
( )T T A A
Properties of transpose of a matrix:
( )T T T A B A B
( )T T TAB B A
( )T T T TABC C B A
( )T T T T A B C A B C
( )T T A A
0 000 0
0 000 0
0 00
0 0 0
Special Matrices
, ( ) A 0 A A A 0
2 0 0 0
1 6 0 0
8 9 3 0
4 2 1 5
-
L
1 2 3 4
0 5 6 7
0 0 8 9
0 0 0 1
U
1) Zero matrix 0:
2) Triangular matrix:
upper triangular matrix lower triangular matrix
12
7 0 0
0 0
0 0 1
D
1 0 0
0 1 0
0 0 1
I
5 0
0 5
3) Diagonal matrix:
4) Scalar matrix:
5) Identity matrix:
1 2 7
2 5 6
7 6 4
A
0 1 1
1 0 2
1 2 0
A
6) Symmetric matrix:
7) Skew symmetric:
;T A A
;T A A
11 21
12 22
a a
aIf
a
A
Determinant of a matrix
(1) Determinant of Matrix): 2 2
11 22 21 12 detThen A a a a a A
11 21 31
12 22 32
13 23 33
a a a
a a a
a a
If
a
A
(2) Determinant of Matrix): 3 3
22 23 21 23 21 2211 21 13
32 33 31 33 31 32
Thena a a a a a
A a a aa a a a a a
Example
Find of the following matrices:
4 3
2 1 i
A
Solution
A
1 2 2
2 5 1
4
5 3
ii
A
4 3
( 4)(1) ( 3)(2) 22 1
i
A
1 2 25 1 2 1 2 5
2 5 1 ( 1) (2) ( 2)5 3 4 3 4 5
4 5 3
ii
A
10 (2) 2 ( 2) 10 14
11 12 1
21 22 2
1 2
,
n
n
n n nn
a a a
a a a
a a a
If
A
Inverse of a matrix
1 1 en Th A adj A
A
T
ij ijadj A adj a A
where:
1A
is the cofactor of each element.ijA
Example
Find of the following matrix:
Solution
1A
1 2 2
2 5 1
4 5
3
A
1 2 2
2 5 1 14
4 5 3
A
T10 2 10
4 5 3
8 3 1
adj A
10 4 8
2 5 3
10 3 1
1
10 4 81
2 5 314
10 3 1
A
* We will solve Linear system of equations which take
the following form:
Solution of linear system of equations
by using Inverse matrix method
11 1 12 2 1 1
21 1 22 2 2 2
1 1 2 2
n n
n n
n n nn n n
a x a x a x b
a x a x a x b
a x a x a x b
to get the values of the unknowns 1 2, , , .nx x x
* We can write the previous system of equations in the
general form as:
AX B
where:
11 12 1 1 1
21 22 2 2 2
1 2
, ,
n
n
n n nn n n
a a a x b
a a a x bX B
a a a x b
A
* To solve this system, multiply both sides by we get:1A
1 1A AX A B 1X A B
Example
Solve the following linear system of equations using
inverse matrix method:
Solution
1 2 3
1 2 3
1 2 3
x 2 2 4
2x 5 7
4x 5 3 5
x x
x x
x x
1 2 2
2 5 1
4 5
3
A1
10 4 81
2 5 314
10 3 1
A
1X A B
1
2
3
10 4 8 41
2 5 3 714
10 3 1 5
x
x
x
40 28 401
8 35 1514
40 21 5
28 21
42 314
456
![jur.mac.edu.pljur.mac.edu.pl/BiF/firstac19.pdf · 2019. 10. 28. · Bargielski Hubert ZESTAW 1. 1. Niech A = [−2;6), B = (−∞;2], C = (−5;2]. Wyzna-czyć A∩B, A\C, B \A,](https://static.fdocuments.pl/doc/165x107/6108ba152a86ac156e37cc68/jurmacedupljurmaceduplbif-2019-10-28-bargielski-hubert-zestaw-1.jpg)
