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8/3/2019 Lesson 1 Linear Algebra
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ENGINEERINGENGINEERINGTECHNOLOGYTECHNOLOGY
MATHEMATICS 1MATHEMATICS 1
FKB10103FKB10103MDM WAN SULIZA WAN HUSAINMDM WAN SULIZA WAN HUSAIN
ROOM M008 ( NEAR CIMB ATM )ROOM M008 ( NEAR CIMB ATM )HP NUMBER :019HP NUMBER :019--21978082197808
8/3/2019 Lesson 1 Linear Algebra
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LESSON 1
LINEAR ALGEBRA
8/3/2019 Lesson 1 Linear Algebra
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3
Matrix Algebra
What is matrix?
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In this lecture, we will lear n about the tools for
solving linear systems of equations
There are several methods we have lear ned beforesuch as solving by graphical method,by using
substitution and elimination method
and etc.
8/3/2019 Lesson 1 Linear Algebra
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Content
Matrix
Determinant
± Diagonal Method ± Cofactors Method
Elementary Row Operations
In
verse Matrix
8/3/2019 Lesson 1 Linear Algebra
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¹¹¹
¹¹
º
¸
©©©
©©
ª
¨
!v
mnm2m1
2n2221
1n1211
nm
aaa
aaa
aaa
A
.
/.//
-
.
What is Matrix A rectangular array of real (or complex)
numbers arranged in m rows and n columns
Dimension
of the matrix
A
Column
Row
8/3/2019 Lesson 1 Linear Algebra
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Some Special MatricesSquare matrix
¹¹
º
¸©©
ª
¨ !
1.70.4
2.61.2D
¹¹¹
º
¸
©©©
ª
¨
!
2105.0
413.4
15.10
E
¹¹¹
º
¸
©©©
ª
¨!
100
010
001
I3
Identity matrix, nI
¹¹ º
¸©©ª
¨!
10
01I2
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8
Basic oper ations of matrix
(+, -, x, T)
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9
Matrix Addition
A is a m x n matrix and B is a m x n matrix
C = A + B
where for all i=1,2,«,m ; j=1,2,«,nijijij bac !
¹¹¹
º
¸
©©©
ª
¨
!
333231
232221
131211
aaa
aaa
aaa
A
¹¹¹
º
¸
©©©
ª
¨
!
333231
232221
131211
bbb
bbb
bbb
B
¹¹¹
º
¸
©©©
ª
¨
!¹¹¹
º
¸
©©©
ª
¨
!
333332323131
232322222121
131312121111
333231
232221
131211
bababa
bababa
bababa
ccc
ccc
ccc
C
8/3/2019 Lesson 1 Linear Algebra
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10
Example
Notice that matrices with different dimension cannot
be added together:
No valid result for A + B
¹¹ º
¸©©ª
¨
!
420
113 A
¹¹ º
¸©©ª
¨
!
420
113 A ¹¹
º
¸©©ª
¨
!
6310
162 B
¹¹ º ¸©©
ª¨
!¹¹
º ¸©©
ª¨
!
10510
075
6432100
116123 B A
¹¹ º
¸©©ª
¨
!
22
13 B
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11
Matrix Subtr action
A is a m x n matrix and B is a m x n matrix
C = A - B
where for all i=1,2,«,m ; j=1,2,«,nijijij bac !
¹¹¹
º
¸
©©©
ª
¨
!
333231
232221
131211
aaa
aaa
aaa
A
¹¹¹
º
¸
©©©
ª
¨
!
333231
232221
131211
bbb
bbb
bbb
B
¹¹¹
º
¸
©©©
ª
¨
!¹¹¹
º
¸
©©©
ª
¨
!
333332323131
232322222121
131312121111
333231
232221
131211
bababa
bababa
bababa
ccc
ccc
ccc
C
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12
Example
Notice that matrices with different dimension cannot
be subtracted together:
No valid result for A - B
¹¹ º
¸©©ª
¨
!
420
113 A
¹¹ º
¸©©ª
¨
!
420
113 A ¹¹
º
¸©©ª
¨
!
6310
162 B
¹¹ º ¸©©
ª¨
!¹¹
º ¸©©
ª¨
!
2110
251
6432100
116123
)( B A
¹¹ º
¸©©ª
¨
!
22
13 B
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13
Scalar Multiplication
A is a m x n matrix and E is a scalar (real number)
C = E A
where for all i=1,2,«,m ; j=1,2,«,n
e.g. C = A
ijij ac vE!
¹¹¹
º
¸
©©©
ª
¨
vEvEvE
vEvEvE
vEvEvE
!¹¹¹
º
¸
©©©
ª
¨
!
333231
232221
131211
333231
232221
131211
aaa
aaa
aaa
ccc
ccc
ccc
C
¹¹¹
º
¸
©©©
ª
¨
!¹¹¹
º
¸
©©©
ª
¨
vvv
vvv
vvv
!
987
654
321
901080107010
601050104010
301020101010
...
...
...
C ¹¹¹
º
¸
©©©
ª
¨
!
908070
605040
302010
AE !
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14
Matrix Multiplication
A is a m x n matrix and B is a n x p matrix
C = A X B ,where C is a m x p matrix
Where
for all i=1,2,«,m ; j=1,2,«,p
To find the element cij (i-th row and j-th column of C = AB),we have to multiply each element in the i-th row of A by thecorresponding element in the j-th column of B and addthem together.
§!
!!n
k
k jik n jin ji jiij babababac1
2211 .
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15
Matrix Multiplication
C11 from 1st row of A and 1st column of B
[a11 a12 a13«a1m] 1st row of A
[b11 b21 b31«bm1] 1st column of B
C11= a11 b11 + a12 b21 + a13 b31 +«+ a1m bm1
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16
Example
A is a 2 x 3 matrix and B is a 3 x 3 matrix
C = A X B, expect C is a 2 x 3 matrix
Find determine C, elements by elements
C11= a11 b11 + a12 b21 + a13 b31
= 1x1 + 2x0 + 3x(-1)
= -2
¹¹ º
¸©©ª
¨!
123
321 A
¹¹¹
º
¸
©©©
ª
¨
!
231
120
201
B ¹¹ º
¸©©ª
¨!!
232221
131211
ccc
ccc ABC
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17
Example
C21= a21 b11 + a22 b21 + a23 b31 = 3x1 + 2x0 + 1x(-1)
= 2
C12= a11 b12 + a12 b22 + a13 b32 = 1x0 + 2x2 + 3x3
= 13
¹¹ º
¸©©ª
¨!
123
321 A
¹¹¹
º
¸
©©©
ª
¨
!
231
120
201
B ¹¹ º
¸©©ª
¨!!
232221
131211
ccc
ccc ABC
¹¹ º
¸
©©ª
¨
! 123
321 A
¹¹
¹
º
¸
©©
©
ª
¨
!231120
201
B ¹¹ º
¸
©©ª
¨
!! 232221
131211
ccc
ccc
ABC
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18
Example
C22= a21 b12 + a22 b22 + a23 b32 = 3x0 + 2x2 + 1x3
= 7
C13= a11 b13 + a12 b23 + a13 b33 = 1x(-2) + 2x1 + 3x2
= 6
¹¹ º
¸©©ª
¨!
123
321 A
¹¹¹
º
¸
©©©
ª
¨
!
231
120
201
B ¹¹ º
¸©©ª
¨!!
232221
131211
ccc
ccc ABC
¹¹ º
¸
©©ª
¨
! 123
321 A
¹¹
¹
º
¸
©©
©
ª
¨
!231120
201
B ¹¹ º
¸
©©ª
¨
!! 232221
131211
ccc
ccc
ABC
8/3/2019 Lesson 1 Linear Algebra
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19
Example
C23= a21 b13 + a22 b23 + a23 b33 = 3x(-2) + 2x1 + 1x2
= -2
Plug in the computed elements back to C, we have
¹¹ º
¸©©ª
¨!
123
321 A
¹¹¹
º
¸
©©©
ª
¨
!
231
120
201
B ¹¹ º
¸©©ª
¨!!
232221
131211
ccc
ccc ABC
¹¹ º
¸©©ª
¨
!!
272
6132 ABC
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20
Example
What is AxB?
What is BxA?
¹¹ º
¸©©ª
¨!¹¹
º
¸©©ª
¨
vvvv
vvvv!
01
00
01001100
00001000 AB
¹¹ º
¸©©ª
¨!
10
00 A ¹¹
º
¸©©ª
¨!
01
00 B
¹¹ º
¸©©ª
¨!¹¹
º
¸©©ª
¨
vvvv
vvvv!
00
00
10010001
10000000 BA
A B B A v{vMultiplicatio
norder is importa
nt!
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21
Tr anspose
The transpose of a m x n matrix A, denoted
by = B, is the n x m matrix where
E.g.
jiij ab !
TA
¹¹ º
¸©©ª
¨! 654
321A
¹¹
¹
º
¸
©©
©
ª
¨
!63
52
41T
A
321!B
¹
¹¹
º
¸
©
©©
ª
¨
!
3
2
1T
B
8/3/2019 Lesson 1 Linear Algebra
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DeterminantsIf A is square matrix then the determinant
function is denoted by det and det(A) is defined
to be the sum of all the signed elementary
matrices of A.
nnn2n1
2n2221
1n1211
aaa
aaa
aaa
.
/.//
-
.
The result of a
determinant isa singlenumber.
!! AAdet
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Finding Determinants
³D
ia
gona
l Meth
od´Determinant function for a 2×2 matrix.
2221
1211
aaaa!! AAdet
2211aa!
1221aa
8/3/2019 Lesson 1 Linear Algebra
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Example 1
Compute the det(A) for the following matrix.
¹¹ º
¸
©©ª
¨
! 24
53
A
24
53Adet ! 26! 206 !
8/3/2019 Lesson 1 Linear Algebra
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Finding Determinants
³D
ia
gona
l Meth
od´Determinant function for a 3×3 matrix.
!B
332211aaa!
132231aaa
333231
232221
131211
aaa
aaa
aaa
3231
2221
1211
aa
aa
aa
312312aaa
322113aaa
112332aaa
122133aaa
8/3/2019 Lesson 1 Linear Algebra
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Example 2
Compute the det(A) for the following matrix.
¹¹¹
º
¸
©©©
ª
¨
!
111
420
022
A
8/3/2019 Lesson 1 Linear Algebra
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Example 2«solution
!B
4! 0
1-11
4-2-0
02-2
11
2-0
2-2
8 0 8- 0!B 20
8/3/2019 Lesson 1 Linear Algebra
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Minor & Cof actor If A is a square matrix then the minor of ,
denoted by , is the determinant of the
submatrix that results from removing the i th row
and j th column of A.
jia
ji
ji
ji M1C
!
jiM
If A is a square matrix then the cof actor of ,
denoted by , is the number
jia
jiC
8/3/2019 Lesson 1 Linear Algebra
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Example 3
For the following matrix compute the minor, M23
and the cofactor, C23
¹¹¹
º
¸
©©©
ª
¨
!
740
259
613
A
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Example 3«solution
!740259
613
M 32
! 40
13
!32C
12
!
321
32
M1 12
12!
Minor
Cofactor
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Finding Determinants
³Meth
od of Cof a
ctors´
nnn2n1
2n2221
1n1211
aaa
aaa
aaa
.
/.//
-
.
!A
If A is an n×n matrix.
Choose any row, then:
nini2i2i1i1i Ca.....CaCa !A
Choose any column, then:
jn jn j2 j2 j1 j1 Ca.....CaCa !A
cof actor
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Example 4
Compute the det(A) for the following matrix.
¹¹¹
º
¸
©©©
ª
¨
!
111
420
022
A
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Example 4«solution
2!1142
04262 !
Choose first row, then:
312111 C0C2C2 !A 312111 M0M-2M2 !
20A !
0
¹¹¹
º
¸
©©©
ª
¨
!
111
420
022
A
2
11
40
8/3/2019 Lesson 1 Linear Algebra
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Elementary Row Oper ations
Elementary row operations are manipulations
that can be performed on the row of a matrix.
¹¹ º
¸©©ª
¨
43
21
Three basic elementary operations.
1. Interchange any two rows.1. Interchange any two rows.
¹¹ º
¸©©ª
¨
21
4321 R R !
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Elementary Row Oper ations
¹¹ º ¸©©
ª¨
4321
2. Multiply any row by a non2. Multiply any row by a non--zerozero
scalar.scalar.
¹¹ º ¸©©
ª¨
434211
R 2R !
¹¹ º
¸©©ª
¨
43
21
3. Add k times of any row to another 3. Add k times of any row to another
row.row.
¹¹ º
¸©©ª
¨
43
85211 R R 2R !
8/3/2019 Lesson 1 Linear Algebra
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Example 5
Use elementary row operation to transform
¹¹ º
¸©©ª
¨
42
31into ¹¹
º
¸©©ª
¨
10
01
8/3/2019 Lesson 1 Linear Algebra
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¹¹
º
¸©©
ª
¨ 31
Example 5«solution
¹¹ º
¸©©ª
¨ 42
31
122 2R R R! ¹¹ º
¸
©©ª
¨ 31
100
10
10062:2R
42:R
1
2
¹¹ º
¸©©ª
¨
10R R 22
z!10
10
10
10
0:
10
R 2
211 3R R R !
01
30:3
31:
2
1
R
R
01
10
8/3/2019 Lesson 1 Linear Algebra
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Inverse Matrix
If A is square matrix and we can find
another matrix of the same size B, such
that
Then we call A invertible and we say that
B is an inverse of the matrix A
nIABBA !!
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Remarks on Inverse Matrix
Inverse matrix is only for square matrix
The inverse for a matrix is unique
The inverse of a matrix A is denoted as A-1
If A is invertible, then A is a non-singular
matrix
If A is not invertible, then A is a singular matrix
8/3/2019 Lesson 1 Linear Algebra
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¹¹ º
¸
©©ª
¨
Finding Inverse Matrix
³Using Row Oper a
tion´Inverse for a 2×2 matrix.
operation
r owelementary
Find row operations that will convert the
first 2 columns into I2.
1
A
2I
The third and fourth columns should then
contain A-1.
¹¹ º
¸
©©ª
¨22A v 2I
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Example 6
Determine the inverse of matrix A.
¹¹ º
¸©©ª
¨
!
42
31A
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Example 6«solution
Form a new matrix
¹
¹
º
¸
©
©
ª
¨
10
01
42
31Matrix
A
122 2R R R! ¹¹
º
¸©©
ª
¨ 0131
12100
12-100
0262:2R
1042:R
1
2
10
R
22R
! ¹¹ º
¸©©ª
¨ 0131
10
1
5
110 101
51
101
102
1010
10-0
10
R
10
:2
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Example 6«solution
¹¹ º
¸©©ª
¨
10
1
5
110
¹¹ º
¸©©ª
¨
!
101
51
103
52
1A
211 3R R R !01
10
3
5
2
103
52
103
53
2
1
01
30:3R
0131:R
The inverse of matrix A is
8/3/2019 Lesson 1 Linear Algebra
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¹¹ º
¸
©©ª
¨
Finding Inverse Matrix³Using Row Oper ation´
Inverse for a 3×3 matrix.
operation
r owelementary
Find row operations that will convert the
first 3 columns into I3.
1
A
3I
The last three columns should then contain
A-1.
¹¹ º
¸
©©ª
¨33A v 3I
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Finding Inverse Matrix³Using Row Oper ation´
Please refer the attachment
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Finding Inverse Matrix³Using Adjoint Cof actor´
Inverse for a 2×2 matrix.
If matrix
is invertible, its inverse will be
¹¹ º ¸©©
ª¨!
dc baA
¹¹ º
¸©©ª
¨
!
ac
bd
A
1A 1
8/3/2019 Lesson 1 Linear Algebra
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Example 7
Determine the inverse for the following
matrix.
¹¹ º
¸
©©ª
¨
! 24
53
A
¹¹ º ¸©©
ª¨v!1
A
45
26 32
A
1¹¹ º ¸©©
ª¨
!26
3
13
2
26
5
13
1
8/3/2019 Lesson 1 Linear Algebra
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Finding Inverse Matrix³Using Adjoint Cof actor´
Inverse for a 3×3 matrix.
If matrix
is in
vertible, its inverse will be
¹¹¹
º
¸
©©©
ª
¨
!
333231
232221
131211
A
aaa
aaa
aaa
Aad jointA
1A
1 v!
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For Example please refer the attachment