Chapter 1 Matrices and Systems of Equations. 1Systems of Linear Equations Where the a ij ’s and b...
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Transcript of Chapter 1 Matrices and Systems of Equations. 1Systems of Linear Equations Where the a ij ’s and b...
![Page 1: Chapter 1 Matrices and Systems of Equations. 1Systems of Linear Equations Where the a ij ’s and b i ’s are all real numbers, x i ’s are variables. We.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649d4d5503460f94a2c098/html5/thumbnails/1.jpg)
Chapter 1
Matrices and Systems of Equations
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1 Systems of Linear Equations
)1(
2211
22222121
11212111
mnmnmm
nn
nn
bxaxaxa
bxaxaxa
bxaxaxa
Where the aij’s and bi’s are all real numbers, xi’s are variables . We will refer to systems of the form (1) as m×n linear systems.
![Page 3: Chapter 1 Matrices and Systems of Equations. 1Systems of Linear Equations Where the a ij ’s and b i ’s are all real numbers, x i ’s are variables. We.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649d4d5503460f94a2c098/html5/thumbnails/3.jpg)
Definition Inconsistent : A linear system has no solution.Consistent : A linear system has at least one solution.
Example( ) xⅰ 1 + x2 = 2 x1 − x2 = 2
( ) ⅱ x1 + x2 = 2 x1 + x2 =1
( ) ⅲ x1 + x2 = 2 −x1 − x2 =-2
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Definition Two systems of equations involving the same variables are said to be equivalent if they have the same solution set.
Three Operations that can be used on a system to obtain an equivalent system:
Ⅰ. The order in which any two equations are written may be interchanged.
Ⅱ. Both sides of an equation may be multiplied by the same nonzero real number.
Ⅲ. A multiple of one equation may be added to (or subtracted from) another.
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n×n Systems
Definition
A system is said to be in strict triangular form if in the kthequation the coefficients of the first k-1 variables are all zero and the coefficient of xk is nonzero (k=1, …,n).
42
2
123
3
32
321
x
xx
xxx
is in strict triangular form.
Example The system
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Example Solve the system
1
432
33
32
321
321
321
xxx
xxx
xxx
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Elementary Row Operations:Ⅰ. Interchange two rows.Ⅱ. Multiply a row by a nonzero real number.Ⅲ. Replace a row by its sum with a multiple of another row.
Example Solve the system
3223
1242
6
0
4321
4321
4321
432
xxxx
xxxx
xxxx
xxx
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2 Row Echelon Form
1
1
1
1
1
42211
31100
30022
10011
11111
0
1
3
0
1
31100
31100
52200
21100
11111
0
1
3
0
1
10000
10000
10000
21100
11111
3
4
3
0
1
00000
00000
10000
21100
11111
pivotal row
pivotal row
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Definition
A matrix is said to be in row echelon form
ⅰ. If the first nonzero entry in each nonzero row is 1.
ⅱ. If row k does not consist entirely of zeros, the numb
er of leading zero entries in row k+1 is greater than the
number of leading zero entries in row k.
ⅲ. If there are rows whose entries are all zero, they are
below the rows having nonzero entries.
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Example Determine whether the following matrices arein row echelon form or not.
010
000
000
100
321
400
530
642
100
310
241
01
10
0000
3100
0131
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Definition
The process of using operations , , Ⅰ Ⅱ Ⅲ to transform a l
inear system into one whose augmented matrix is in ro
w echelon form is called Gaussian elimination.
Definition
A linear system is said to be overdetermined if there ar
e more equations than unknows.
A system of m linear equations in n unknows is said to
be underdetermined if there are fewer equations than u
nknows (m<n).
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Example
232
322
2
)(
22
3
1
)(
54321
54321
54321
21
21
21
xxxxx
xxxxx
xxxxx
b
xx
xx
xx
a
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Definition
A matrix is said to be in reduced row echelon form if:
ⅰ. The matrix is in row echelon form.
ⅱ. The first nonzero entry in each row is the only nonz
ero entry in its column.
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Homogeneous SystemsA system of linear equations is said to be homogeneous if the constants on the right-hand side are all zero.
Theorem 1.2.1 An m×n homogeneous system of linear equations has a nontrivial solution if n>m.
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3 Matrix Algebra
Matrix Notation
mnmm
n
n
aaa
aaa
aaa
A
21
22221
11211
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VectorsVectors
row vector row vector
column vector column vector
nxxxX ,,, 21 1×n matrix matrix
nx
x
x
X2
1
n×1 matrix matrix
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Definition
Two m×n matrices A and B are said to be equal if aij=bij f
or each i and j.
Scalar MultiplicationIf A is a matrix and k is a scalar, then kA is the matrix
formed by multiplying each of the entries of A by k.
Definition
If A is an m×n matrix and k is a scalar, then kA is the m×
n matrix whose (i, j) entry is kaij.
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Matrix AdditionTwo matrices with the same dimensions can be added
by adding their corresponding entries.
Definition
If A=(aij) and B=(bij) are both m×n matrices,then the sum
A+B is the m×n matrix whose (i, j) entry is aij+bij for eac
h ordered pair (i, j).
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1 1 2 1 0 02 0 3 , 0 1 0
1 1 2 0 0 1A I
2 3A I
Example
Let
Then calculate 。
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Matrix Multiplication
Definition
If A=(aij) is an m×n matrix and B=(bij) is an n×r matrix, th
en the product AB=C=(cij) is the m×r matrix whose entri
es are defined by
ccijij = = aaii11bb11jj + + aaii22bb22jj +…+ +…+ aaiinnbbnnjj = = aaiikkbbkkjj. .
kk=1=1
nn
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Example
,021114
,012301
BA then calculate AB.1. If
2. If ,1111,22
11
BA then calculate AB and BA.
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Matrix Multiplication and Linear SystemsCase 1 One equation in Several Unknows
If we let and
then we define the product AX by
)( 21 naaaA
nx
x
x
X2
1
nnxaxaxaAX 2211
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Case 2 M equations in N Unknows
If we let and
then we define the product AX by
mnmm
n
n
aaa
aaa
aaa
A
21
22221
11211
nx
x
x
X2
1
nmnmm
nn
nn
xaxaxa
xaxaxa
xaxaxa
AX
2211
2222121
1212111
![Page 24: Chapter 1 Matrices and Systems of Equations. 1Systems of Linear Equations Where the a ij ’s and b i ’s are all real numbers, x i ’s are variables. We.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649d4d5503460f94a2c098/html5/thumbnails/24.jpg)
Definition
If a1, a2, … , an are vectors in Rm and c1, c2, … , cn are scalars,
then a sum of the form
c1a1+c2a2+‥‥cnan
is said to be a linear combination of the vectors a1, a2, … , an .
Theorem 1.3.1 (Consistency Theorem for Linear
Systems)
A linear system AX=b is consistent if and only if b can be
written as a linear combination of the column vectors of
A.
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Theorem 1.3.2 Each of the following statements is vali
d for any scalars k and l and for any matrices A, B and
C for which the indicated operations are defined.
1. A+B=B+A
2. (A+B)+C=A+(B+C)
3. (AB)C=A(BC)
4. A(B+C)=AB+AC
5. (A+B)+C=AC+BC
6. (kl)A=k(lA)
7. k(AB)=(kA)B=A(kB)
8. (k+l)A=kA+lA
9. k(A+B)=kA+kB
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The Identity Matrix
Definition
The n×n identity is the matrix where)( ijI
jiif
jiifij 0
1
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Matrix Inversion
Definition
An n×n matrix A is said to be nonsingular or invertible if
there exists a matrix B such that AB=BA=I.
Then matrix B is said to be a multiplicative inverse of A.
Definition
An n×n matrix is said to be singular if it does not have a
multiplicative inverse.
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Theorem 1.3.3 If A and B are nonsingular n×n matrices,
then AB is also nonsingular and (AB)-1=B-1A-1
The Transpose of a Matrix
Definition
The transpose of an m×n matrix A is the n×m matrix B d
efined by
bji=aij
for j=1, …, n and i=1, …, m. The transpose of A is denote
d by AT.
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Algebra Rules for Transpose:1. (AT)T=A
2. (kA)T=kAT
3. (A+B)T=AT+BT
4. (AB)T=BTAT
Definition
An n×n matrix A is said to be symmetric if AT=A.
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4. Elementary Matrices
If we start with the identity matrix I and then perform
exactly one elementary row operation, the resulting matrix
is called an elementary matrix.
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Type I. An elementary matrix of type I is a matrix obtained by interchanging two rows of I.
Example Let
100
001
010
1E and let A be a 3×3 matrix
then
333231
131211
232221
333231
232221
131211
1
100
001
010
aaa
aaa
aaa
aaa
aaa
aaa
AE
333132
232122
131112
333231
232221
131211
1
100
001
010
aaa
aaa
aaa
aaa
aaa
aaa
AE
![Page 32: Chapter 1 Matrices and Systems of Equations. 1Systems of Linear Equations Where the a ij ’s and b i ’s are all real numbers, x i ’s are variables. We.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649d4d5503460f94a2c098/html5/thumbnails/32.jpg)
Type II. An elementary matrix of type II is a matrix obtained by multiplying a row of I by a nonzero constant.
Example Let
300
010
001
2E and let A be a 3×3 matrix
then
333231
232221
131211
333231
232221
131211
2
333300
010
001
aaa
aaa
aaa
aaa
aaa
aaa
AE
333231
232221
131211
333231
232221
131211
2
3
3
3
300
010
001
aaa
aaa
aaa
aaa
aaa
aaa
AE
![Page 33: Chapter 1 Matrices and Systems of Equations. 1Systems of Linear Equations Where the a ij ’s and b i ’s are all real numbers, x i ’s are variables. We.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649d4d5503460f94a2c098/html5/thumbnails/33.jpg)
Type III. An elementary matrix of type III is a matrix obtained from I by adding a multiple of one row to another row.
Example Let
100
010
301
3E and let A be a 3×3 matrix
333231
232221
331332123111
333231
232221
131211
3
333
100
010
301
aaa
aaa
aaaaaa
aaa
aaa
aaa
AE
33313231
23212221
13111211
333231
232221
131211
3
3
3
3
100
010
301
aaaa
aaaa
aaaa
aaa
aaa
aaa
AE
![Page 34: Chapter 1 Matrices and Systems of Equations. 1Systems of Linear Equations Where the a ij ’s and b i ’s are all real numbers, x i ’s are variables. We.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649d4d5503460f94a2c098/html5/thumbnails/34.jpg)
In general, suppose that E is an n×n elementary matri
x. E is obtained by either a row operation or a column op
eration.
If A is an n×r matrix, premultiplying A by E has the
effect of performing that same row operation on A. If B
is an m×n matrix, postmultiplying B by E is equivalent
to performing that same column operation on B.
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Example
Let ,
Find the elementary matrices , , such that .1P 2P 1 2B PAP
11 12 13
21 22 23
31 32 33
a a aA a a a
a a a
31 32 33 33
21 22 23 23
11 12 13 13
333
a a a aB a a a a
a a a a
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Theorem 1.4.1 If E is an elementary matrix, then E is
nonsingular and E-1 is an elementary matrix of the
same type.
Definition
A matrix B is row equivalent to A if there exists a finite
sequence E1, E2, … , Ek of elementary matrices such that
B=EkEk-1‥‥E1A
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Theorem 1.4.2 (Equivalent Conditions for Nonsingularity)
Let A be an n×n matrix. The following are equivalent:
(a) A is nonsingular.
(b) Ax=0 has only the trivial solution 0.
(c) A is row equivalent to I.
Theorem 1.4.3 The system of n linear equations in n
unknowns Ax=b has a unique solution if and only if A
is nonsingular.
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If A is nonsingular, then A is row equivalent to I and
hence there exist elementary matrices E1, …, Ek such
that
EkEk-1‥‥E1A=I multiplying both sides of this
equation on the right by A-1
EkEk-1‥‥E1I=A-1
Thus (A I) (I A-1) row operations
A method for finding the inverse of a matrix
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Example Compute A-1 if
322
021
341
A
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Example Solve the system
8322
122
1234
321
21
321
xxx
xx
xxx
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Diagonal and Triangular Matrices
An n×n matrix A is said to be upper triangular if aij=0 for i>j and lower triangular if aij=0 for i<j.
An n×n matrix A is said to be diagonal if aij=0 whenever i≠j .
A is said to be triangular if it is either upper triangular or lower triangular.
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5. Partitioned Matrices
C=
1 -2 4 1 3
2 1 1 1 1
3 3 2 -1 2
4 6 2 2 4
C11 C12
= C21 C22
-1 2 1
B= 2 3 1
1 4 1
=(b1, b2, b3)
AB=A(b1, b2, b3)=(Ab1, Ab2, Ab3)
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In general, if A is an m×n matrix and B is an n×r that has
been partitioned into columns (b1, …, br), then the block
multiplication of A times B is given by
AB=(Ab1, Ab2, … , Abr)
If we partition A into rows, then
:),(
:),2(
:),1(
ma
a
a
A
Then the product AB can be partitioned into rows as follows:
Bma
Ba
Ba
AB
:),(
:),2(
:),1(
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Block Multiplication
Let A be an m×n matrix and B an n×r matrix.
Case 1 B=(B1 B2), where B1 is an n×t matrix and B2
is an n×(r-t) matrix.
AB= A(b1, … , bt, bt+1, … , br) = (Ab1, … , Abt, Abt+1, … , Abr) = (A(b1, … , bt), A(bt+1, … , br)) = (AB1 AB2)
![Page 45: Chapter 1 Matrices and Systems of Equations. 1Systems of Linear Equations Where the a ij ’s and b i ’s are all real numbers, x i ’s are variables. We.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649d4d5503460f94a2c098/html5/thumbnails/45.jpg)
Case 2 A= ,where A1 is a k×n matrix and A2
is an (m-k)×n matrix.
2
1
A
A
Thus
BA
BAB
A
A
2
1
2
1
Case 3 A=(A1 A2) and B= , where A1 is an m×s matrix
and A2 is an m×(n-s) matrix, B1 is an s×r matrix and B2 is an
(n-s)×r matrix.
2
1
B
B
Thus 22112
121 BABAB
BAA
![Page 46: Chapter 1 Matrices and Systems of Equations. 1Systems of Linear Equations Where the a ij ’s and b i ’s are all real numbers, x i ’s are variables. We.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649d4d5503460f94a2c098/html5/thumbnails/46.jpg)
Case 4 Let A and B both be partitioned as follows :
A11 A12 kA= A21 A22 m-k
s n-s
B11 B12 sB= B21 B22 n-s
t r-t
2222122121221121
2212121121121111
2221
1211
2221
1211
BABABABA
BABABABA
BB
BB
AA
AA
Then
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In general, if the blocks have the proper
dimensions, the block multiplication can be
carried out in the same manner as ordinary
matrix multiplication.
![Page 48: Chapter 1 Matrices and Systems of Equations. 1Systems of Linear Equations Where the a ij ’s and b i ’s are all real numbers, x i ’s are variables. We.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649d4d5503460f94a2c098/html5/thumbnails/48.jpg)
,
000100311132
401
,
040006110023010
52001
BA
Example
Let
Then calculate AB.
![Page 49: Chapter 1 Matrices and Systems of Equations. 1Systems of Linear Equations Where the a ij ’s and b i ’s are all real numbers, x i ’s are variables. We.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649d4d5503460f94a2c098/html5/thumbnails/49.jpg)
Example
Let A be an n×n matrix of the form
22
11
0
0
A
A
where A11 is a k×k matrix (k<n). Show that
A is nonsingular if and only if A11 and A22
are nonsingular.