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Matrices
&
Linear Equations
Chapter One
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Definition
• A matrix is a rectangular array of numbers.
The numbers in the array are called the
entries in the matrix.
• A matrix with size (order) m n is a matrix
with m rows and n columns.
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Notation
• An entry that occur in row i and column j of a
matrix A will be denoted by aij
• Thus a general m n matrix might be written
as:
3
mnmmm
n
n
aaaa
aaaa
aaaa
321
2232221
1131211
m rows
n column
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• A matrix can be denoted by an uppercase
letter such as A, B or C . A matrix can also be
denoted by [aij ],[bij ] or [cij ]. Therefore,
4
A = [aij ] =
mnmmm
n
n
aaaa
aaaa
aaaa
321
2232221
1131211
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• A matrix with n rows and n column is called a
square matrix of order n, and the entries a11,
a22, a33, …ann are main diagonal of A;
5
nnnnn
n
n
aaaa
aaaa
aaaa
321
2232221
1131211
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OPERATION OF MATRICES
Two matrices A = [aij] and B = [bij] are defined to be equal if they have thesame size and their corresponding entries are equal (aij = bij for all i and j).
Example 1:
94
32 A ,
9
32
x B ==> A = B only when x = 4
6
• Equal Matrices
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Example 2:
The matrices
540
432121
A
and
z y
xw
B
4
4221
are equal (A = B), if and only if w = -1, x = - 3, y = 0, and z = 5.
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• Addition and Subtraction
If A = [aij] and B = [bij] are matrices of size m n, then their sum A+B isthe m n matrix given by adding the entries of B to the correspondingentries of A i.e.
A + B = [aij + bij].
Their differences A-B is the m n matrix obtained by subtracting theentries of B from the corresponding entries of A i.e.
A B = [aij bij]
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Example 3:Let
0724
4201
3012
A,
5423
1022
1534
B
,
012
321C
Then
5307
3221
4542
B A,
51141
52232526
B A
A+C , B+C , AC , BC are undefined
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If A = [aij] is an m n matrix and c is a scalar, the scalar multiple of A by c is the m n matrix given by
cA = c[aij]
The symbol - A represents the scalar product (1) A. Moreover, if A and B
are of the same size, then A B represents the sum of A and (1) B. That is,
A B = A + (1) B
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• Sclar Multiples
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Example 4:
Let
062
421 A ,
531
720 B ,
1203
369C
11
41914
101629
1203
369 3
531
720
062
421 232 C B A
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Properties
Let A, B and C be m n matrices and let c and d be scalars.
1. A + B = B + A Commutative Property of Matrix Addition
2. A + (B + C) = (A + B) + C Associative Property of Matrix Addition
3. (cd)A = c(dA) Associative Property of Scalar Multiplication
4. 1 A = A Scalar Identity
5. c(A + B) = cA + cB Distributive Property
6. (c + d)A = cA + dA Distributive Property
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Matrix Multiplication
Definition: If A = [aij] is an m n matrix and and B = [bij] is an n p
matrix, the product AB is an m p matrix
AB = [cij] where
njin ji ji ji
n
k
kjik ij bababababac
...332211
1
In order for the product of two matrices to be defined, the number ocolumns of the first matrix must equal the number of rows of the second
matrix.
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Example 5
14
11
13
31.1
cr bqap
r
q
p
cba
12
13
32432
432
4
3
2
.2
f ed
cba
f ed
cba
32
3222
.3
ducr dt cqdscp
buar bt aqbsap AB
ut s
r q p B
d c
ba A
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Properties
Let A, B, and C be matrices and let c be a scalar.
1. A(BC) = (AB)C Associative Property of Multiplication
2. A(B + C)=AB + AC Distributive Property3. (A + B)C = AC + BC Distributive Property
4. c(AB) = (cA)B = A(cB) Distributive Property
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Transpose Matrices
If A is an m n matrix, the transpose matrix T A , is the n m matrix whose
rows are the columns of A in the same order. In other words, the first row oT
A is the first column of A, the second row of T A is the second column of A,
and so on.
Example 6:
685
073
421
A
604
872
531T A
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Properties
Let A and B denotes matrices of the same size, and let k denote a scalar.
1. If A is an m n matrix, then T A is an n m matrix
2. A AT T
3. T T kAkA
4. T T T B A B A
5.T T T A B AB )(
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Symmetric Matrices
A matrix A is called symmetric if A = A
653
592
321
A
653
592
321T A ==> A is a symmetric matrix
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Elementary Row Operations (ERO)
Interchange any two rows (row i and row j) and is denoted as
i R j R
Multiply row i by a scalar k(k 0) and is denoted as
i R ikR
Add multiple of row j to row i and is denoted as
i
R i j
R kR
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a) ERO : Interchange rows 1 and 3
a b c
d e f
g h i
1 R 3 R
g h i
d e f
a b c
b)
ERO : Multiply row 2 by 5, which can be read as
2 R becomes 5 X 2 R
a b c
d e f
g h i
2 R 25 R 5 5 5
a b c
d e f
g h i
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Gaussian Elimination
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Echelon form
A matrix satisfying the following conditions is said to be in reduced row-
echelon form (RREF):
1. If a row does not consist entirely of zeros, then the first nonzero number
in the row is a 1, which is called a leading 1.
2. If there are any rows that consist entirely of zeros, then they are grouped
together at the bottom of the matrix.
3. In any two successive rows that do not consist entirely of zeros, the
leading 1 in the lower row occurs farther to the right than the leading 1
in the higher row.4. Each column that contains a leading 1 has zeros elsewhere
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Example
The following matrices are in reduced row-echelon form:
00
00,
00000
00000
31000
10210
,
100
010001
,
1100
70104001
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How is about these…?
10000
01100
06210
,
000
010
011
,
5100
2610
7341
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25
Example 7: Find x and y using
Gaussian Elimination Method for:
1470
3/ 83/ 21
6
2116
3/ 83/ 21
3/ 1
2116
823
:formmatrixintoitChange
Solution
2116
823
12
1
R R
R
y x
y x
2,4
210
401
3/ 2
210
3/ 83/ 21
7/ 1
21
2
y x
R R
R
Example 8:Find x y and z using
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Example 8:Find x, y and z using
Gaussian Elimination Method for:
26
3/53/53/100
5410
3/43/13/21
)3(
3/53/53/100
3/53/43/10
3/43/13/21
)2(),1(
11223111
3/43/13/21
)3/1(
1122
3111
4123
2
1312
1
R
R R R R
R
1,1,1
1100
1010
1001)3(),4(
1100
5410
2301)15/1(
151500
5410
2301
)3/10(),3/2(
3132
3
2321
z y x
R R R R
R
R R R R
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Inverse of a 2x2 matrix
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More on Invertible Matrices
321
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Example 9:
Using Row Operations to find A-1 for
Solution:
Begin with:
Use successive row operations to produce:
801
352
321
Example 10:
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Example 10:
Linear Systems and Invertible Matrices
From Example 9,