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Applied Linear Algebra
Instructor: Dr. V Ramakrishna Reddy
Open Hours: Friday 04-00 PM to 05-00 PM
Tuesday 13:30 PM to 14:00 PM (Extra Time)
Lecture Set – 01
Chapter 1:
Systems of Linear Equations & Matrices
23 December 2019 Applied Linear Algebra Instructor: Dr.V.R.K.REDDY 1
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Module- 1
Introduction to System of Linear Equations
◼ Introduction to Systems of Linear Equations, Gaussian
Elimination, Matrices and Matrix Operations,
◼ Inverses; Algebraic Properties of Matrices, Elementary
Matrices and a Method for Finding Inverse,
◼ More on Linear Systems and Invertible Matrices
◼ Diagonal, Triangular, and Symmetric Matrices,
◼ Applications of Linear Systems:
❖ Network Analysis (Traffic Flow) and Electrical
Circuits, Leontief Input-Output Models
23 December 2019 Applied Linear Algebra Instructor: Dr.V.R.K.REDDY 2
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23 December 2019 3
Chapter Contents
◼ Introduction to System of Linear Equations
◼ Gaussian Elimination
◼ Matrices and Matrix Operations
◼ Inverses; Rules of Matrix Arithmetic
◼ Elementary Matrices and a Method for Finding A-1
◼ Further Results on Systems of Equations and Invertibility
◼ Diagonal, Triangular, and Symmetric Matrices
Applied Linear Algebra Instructor: Dr.V.R.K.REDDY
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23 December 2019 4
1-1 Linear Equations
◼
Applied Linear Algebra Instructor: Dr.V.R.K.REDDY
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23 December 2019 5
1-1 Example 1 (Linear Equations)
◼,13
2
1,73 ++==+ zxyyx 732 4321 =+−− xxxx
xyxzzyxyx sin and ,423 ,53 ==+−+=+
Applied Linear Algebra Instructor: Dr.V.R.K.REDDY
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23 December 2019 6
1-1 Example 2 (Linear Equations)
◼ Find the solution of x1 – 4x2 + 7x3 = 5
◼ Solution:
❑ We can assign arbitrary values to any two variables and
solve for the third variable
❑ For example
x1 = 5 + 4s – 7t, x2 = s, x3 = t
where s, t are arbitrary values
Applied Linear Algebra Instructor: Dr.V.R.K.REDDY
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23 December 2019 7
1-1 Linear Systems
◼ A finite set of linear equations in the variables x1, x2, …, xn is called a system of linear equations or a linear system.
◼ A sequence of numbers s1, s2, …, sn is called a solution of the system
◼ A system has no solution is said to be inconsistent.
◼ If there is at least one solution of the system, it is called consistent.
◼ Every system of linear equations has either no solutions, exactly one solution, or infinitely many solutions
mnmnmm
nn
nn
bxaxaxa
bxaxaxa
bxaxaxa
=+++
=+++
=+++
...
...
...
2211
22222121
11212111
Applied Linear Algebra Instructor: Dr.V.R.K.REDDY
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◼
23 December 2019 8
1-1 Linear Systems
Applied Linear Algebra Instructor: Dr.V.R.K.REDDY
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23 December 2019
Applied Linear Algebra Instructor:
Dr.V.R.K.REDDY 9
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23 December 2019 Applied Linear Algebra Instructor: Dr.V.R.K.REDDY 10
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23 December 2019 Applied Linear Algebra Instructor: Dr.V.R.K.REDDY 11
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23 December 2019 Applied Linear Algebra Instructor: Dr.V.R.K.REDDY 12
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23 December 2019 Applied Linear Algebra Instructor: Dr.V.R.K.REDDY 13
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23 December 2019 14
1-1 Augmented Matrices
◼ The location of the +s, the xs, and the =s can be abbreviated
by writing only the rectangular array of numbers.
◼ This is called the augmented matrix for the system.
◼ It must be written in the same order in each equation as the
unknowns and the constants must be on the right
mnmnmm
nn
nn
bxaxaxa
bxaxaxa
bxaxaxa
=+++
=+++
=+++
...
...
...
2211
22222121
11212111
mmnmm
n
n
baaa
baaa
baaa
...
...
...
21
222221
111211
1th column
1th row
Applied Linear Algebra Instructor: Dr.V.R.K.REDDY
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23 December 2019 Applied Linear Algebra Instructor: Dr.V.R.K.REDDY 15
Note:
As noted in the introduction to this chapter, the term “matrix” is
used in mathematics to denote a rectangular array of numbers. In a
later section we will study matrices in detail, but for now we will
only be concerned with augmented matrices for linear systems.
For example, the augmented matrix for the system of equations
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23 December 2019 16
1-1 Elementary Row Operations
◼ The basic method for solving a system of linear equations is to
replace the given system by a new system that has the same
solution set but which is easier to solve.
◼ Since the rows of an augmented matrix correspond to the
equations in the associated system, new systems is generally
obtained in a series of steps by applying the following three
types of operations to eliminate unknowns systematically.
mnmnmm
nn
nn
bxaxaxa
bxaxaxa
bxaxaxa
=+++
=+++
=+++
...
...
...
2211
22222121
11212111
mmnmm
n
n
baaa
baaa
baaa
...
...
...
21
222221
111211
Applied Linear Algebra Instructor: Dr.V.R.K.REDDY
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23 December 2019 Applied Linear Algebra Instructor: Dr.V.R.K.REDDY 17
The basic method for solving a linear system is to perform
algebraic operations on the system that do not alter the solution set
and that produce a succession of increasingly simpler systems,
until a point is reached where it can be ascertained whether the
system is consistent, and if so, what its solutions are. Typically, the
algebraic operations are:
1. Multiply an equation through by a nonzero constant.
2. Interchange two equations.
3. Add a constant times one equation to another.
Since the rows (horizontal lines) of an augmented matrix
correspond to the equations in the associated system, these three
operations correspond to the following operations on the rows of
the augmented matrix: (continue)
Method: To Solve the system of algebraic equations
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1-1 Elementary Row Operations
◼ Elementary row operations
❑ Multiply an equation through by an nonzero constant
❑ Interchange two equation
❑ Add a multiple of one equation to another
These are called elementary row operations on a matrix
Applied Linear Algebra Instructor: Dr.V.R.K.REDDY23 December 2019 18
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23 December 2019 19
1-1 Example 3
(Using Elementary Row Operations)
0 563
7 172
9 2
=−+
−=−
=++
zyx
zy
zyx
0563
1342
92
=−+
=−+
=++
zyx
zyx
zyx
−
−
0563
1342
9211
−
−−
0563
17720
9211
0 113
9 2
217
27
=−
−=−
=++
zy
zy
zyx
27113
177 2
9 2
−=−
−=−
=++
zy
zy
zyx
−−
−−
271130
17720
9211
−−
−−
271130
10
9211
217
27
3
9 2
217
27
=
−=−
=++
z
zy
zyx
23
21
217
27
9 2
−=−
−=−
=++
z
zy
zyx
−−
−−
23
21
217
27
00
10
9211
−−
3100
10
9211
2
1727
3
2
1
=
=
=
z
y
x
3
217
27
235
211
=
−=−
=+
z
zy
zx
−−
3100
10
01
2
1727
235
211
3100
2010
1001
Applied Linear Algebra Instructor: Dr.V.R.K.REDDY
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Exercise Problems
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23 December 2019
Applied Linear Algebra Instructor:
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23 December 2019
Applied Linear Algebra Instructor:
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12/23/2019 25
Chapter Contents
◼ Introduction to System of Linear Equations
◼ Gaussian Elimination
◼ Matrices and Matrix Operations
◼ Inverses; Rules of Matrix Arithmetic
◼ Elementary Matrices and a Method for Finding A-1
◼ Further Results on Systems of Equations and Invertibility
◼ Diagonal, Triangular, and Symmetric Matrices
Applied Linear Algebra Instructor: Dr.V.R.K.REDDY23 December 2019 25
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Gaussian Elimination ◼ In this section we will develop a systematic procedure for solving
systems of linear equations.
◼ The procedure is based on the idea of performing certain operations
on the rows of the augmented matrix that simplify it to a form from
which the solution of the system can be ascertained by inspection.
◼ For example, there are many applications that lead to linear systems in
thousands or even millions of unknowns. Large systems require
special techniques to deal with issues of memory size, roundoff errors,
solution time, and so forth. Such techniques are studied in the field of
numerical analysis and will only be touched on in this text. However,
almost all of the methods that are used for large systems are based on
the ideas that we will develop in this section.
23 December 2019 Applied Linear Algebra Instructor: Dr.V.R.K.REDDY 26
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23 December 2019Applied Linear Algebra Instructor: Dr.V.R.K.REDDY
27
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23 December 2019 28
1-2 Echelon Forms
◼ A matrix is in reduced row-echelon form
❑ If a row does not consist entirely of zeros, then the first nonzero number in the row is a 1. We call this a leader 1.
❑ If there are any rows that consist entirely of zeros, then they are grouped together at the bottom of the matrix.
❑ In any two successive rows that do not consist entirely of zeros, the leader 1 in the lower row occurs farther to the right than the leader 1 in the higher row.
❑ Each column that contains a leader 1 has zeros everywhere else.
◼ A matrix that has the first three properties is said to be in row-echelon form.
◼ Note: A matrix in reduced row-echelon form is of necessity in row-echelon form, but not conversely.
3100
2010
1001
Applied Linear Algebra Instructor: Dr.V.R.K.REDDY
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23 December 2019 Applied Linear Algebra Instructor: Dr.V.R.K.REDDY 29
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23 December 2019 30
1-2 Example 2
◼ Matrices in row-echelon form (any real numbers substituted for the
*’s. ) :
◼ Matrices in reduced row-echelon form (any real numbers
substituted for the *’s. ) :
*100000000
*0**100000
*0**010000
*0**001000
*0**000*10
,
0000
0000
**10
**01
,
0000
*100
*010
*001
,
1000
0100
0010
0001
*100000000
****100000
*****10000
******1000
********10
,
0000
0000
**10
***1
,
0000
*100
**10
***1
,
1000
*100
**10
***1
Applied Linear Algebra Instructor: Dr.V.R.K.REDDY
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23 December 2019
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23 December 2019
Applied Linear Algebra Instructor:
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23 December 2019 Applied Linear Algebra Instructor: Dr.V.R.K.REDDY 33
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1-2 Example 3
◼ Solutions of linear systems
4000
2010
5001
1000
0210
0001
−
2
6
1
3100
2010
4001
−
000000
251000
130100
240061
Applied Linear Algebra Instructor: Dr.V.R.K.REDDY23 December 2019 34
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23 December 2019 35
1-2 Elimination Methods
◼ A step-by-step elimination procedure that can be used to
reduce any matrix to reduced row-echelon form
−−−
−
−
156542
281261042
1270200
Applied Linear Algebra Instructor: Dr.V.R.K.REDDY
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23 December 2019 36
1-2 Elimination Methods
◼ Step1. Locate the leftmost column that does not consist entirely of
zeros.
◼ Step2. Interchange the top row with another row, to bring a nonzero
entry to top of the column found in Step1
−−−
−
−
156542
281261042
1270200
Leftmost nonzero column
−−−
−
−
156542
1270200
281261042The 1th and 2th rows in the preceding matrix were interchanged.
−−−
−
−
156542
281261042
1270200
Applied Linear Algebra Instructor: Dr.V.R.K.REDDY
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23 December 2019 37
1-2 Elimination Methods
◼ Step3. If the entry that is now at the top of the column found in
Step1 is a, multiply the first row by 1/a in order to introduce a
leading 1.
◼ Step4. Add suitable multiples of the top row to the rows below so
that all entries below the leading 1 become zeros
−−−
−
−
156542
1270200
1463521
The 1st row of the preceding matrix was multiplied by 1/2.
−−
−
−
29170500
1270200
1463521-2 times the 1st row of the preceding matrix was added to the 3rd row.
Applied Linear Algebra Instructor: Dr.V.R.K.REDDY
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23 December 2019 38
1-2 Elimination Methods
◼ Step5. Now cover the top row in the matrix and begin again with
Step1 applied to the submatrix that remains. Continue in this way
until the entire matrix is in row-echelon form
−−−
−
−
29170500
1270200
1463521
The 1st row in the submatrix was multiplied by -1/2 to introduce a leading 1.
−−
−−
−
29170500
60100
1463521
27
Leftmost nonzero column in the submatrix
Applied Linear Algebra Instructor: Dr.V.R.K.REDDY
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23 December 2019 39
1-2 Elimination Methods
◼ The last matrix is in reduced row-echelon form
−−
−
210000
60100
1463521
27
-5 times the 1st row of the submatrix was added to the 2nd row of the submatrix to introduce a zero below the leading 1.
−−
−
10000
60100
1463521
21
27
−−
−
10000
60100
1463521
21
27
The top row in the submatrix was covered, and we returned again Step1.
The first (and only) row in the new submetrix was multiplied by 2 to introduce a leading 1.
Leftmost nonzero column in the new submatrix
Applied Linear Algebra Instructor: Dr.V.R.K.REDDY
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23 December 2019 40
1-2 Elimination Methods
◼ Step1~Step5: the above procedure produces a row-echelon form and
is called Gaussian elimination
◼ Step1~Step6: the above procedure produces a reduced row-echelon
form and is called Gaussian-Jordan elimination
◼ Every matrix has a unique reduced row-echelon form but a row-
echelon form of a given matrix is not unique
◼ Back-Substitution
❑ To solve a system of linear equations by using Gaussian elimination to
bring the augmented matrix into row-echelon form without continuing
all the way to the reduced row-echelon form.
❑ When this is done, the corresponding system of equations can be solved
by solved by a technique called back-substitution
Applied Linear Algebra Instructor: Dr.V.R.K.REDDY
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1-2 Example 4
◼ Solve by Gauss-Jordan elimination
6 18 48 62
5 15 10 5
1 3 425 62
0 2 2 3
65421
643
654321
5321
=++++
=++
−=−+−−+
=+−+
xxxxx
xxx
xxxxxx
xxxx
Applied Linear Algebra Instructor: Dr.V.R.K.REDDY23 December 2019 41
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1-2 Example 5
◼ From the computations in example 4 , a row-echelon form of the
augmented matrix is given.
◼ To solve the system of equations:
−
0
3/1
1
0
000000
100000
302100
020231
1/3
1 3 2
0 2 2 3
6
643
5321
=
=++
=+−+
x
xxx
xxxx
Applied Linear Algebra Instructor: Dr.V.R.K.REDDY23 December 2019 42
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1-2 Example 6
◼ Solve the system of equations by Gaussian elimination and
back-substitution.
0563
1342
92
=−+
=−+
=++
zyx
zyx
zyx
Applied Linear Algebra Instructor: Dr.V.R.K.REDDY23 December 2019 43
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23 December 2019 44
1-2 Homogeneous Linear Systems◼ A system of linear equations is said to be homogeneous if the constant
terms are all zero.
◼ Every homogeneous system of linear equation is consistent, since all
such system have x1 = 0, x2 = 0, …, xn = 0 as a solution.
❑ This solution is called the trivial solution.
❑ If there are another solutions, they are called nontrivial solutions.
◼ There are only two possibilities for its solutions:
❑ There is only the trivial solution
❑ There are infinitely many solutions in addition to the trivial solution
0...
0 ...
0 ...
2211
2222121
1212111
=+++
=+++
=+++
nmnmm
nn
nn
xaxaxa
xaxaxa
xaxaxa
Applied Linear Algebra Instructor:
Dr.V.R.K.REDDY
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23 December 2019 45
1-2 Example 7
◼ Solve the homogeneous
system of linear equations by
Gauss-Jordan elimination
◼ The augmented matrix
◼ Reducing this matrix to
reduced row-echelon form
◼ The general solution is
◼ Note: the trivial solution isobtained when s = t = 0
0
0 2
0 32
0 2 2
543
5321
54321
5321
=++
=−−+
=+−+−−
=+−+
xxx
xxxx
xxxxx
xxxx
−−
−−−
−
001000
010211
013211
010122
000000
001000
010100
010011
txxtx
sxtsx
==−=
=−−=
543
21
,0,
,
Applied Linear Algebra Instructor:
Dr.V.R.K.REDDY
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23 December 2019 46
1-2 Example 7 (Gauss-Jordan Elimination)
◼ Two important points:
❑ None of the three row operations alters the final column of zeros, so the
system of equations corresponding to the reduced row-echelon form of
the augmented matrix must also be a homogeneous system.
❑ If the given homogeneous system has m equations in n unknowns with
m < n, and there are r nonzero rows in reduced row-echelon form of the
augmented matrix, we will have r < n. It will have the form:
❑ (Theorem 1.2.1)
0()
0()
0()
2
1
=+
=+
=+
kr
k
k
x
x
x
−=
−=
−=
()
()
()
2
1
kr
k
k
x
x
x
mnmnmm
nn
nn
bxaxaxa
bxaxaxa
bxaxaxa
=+++
=+++
=+++
...
...
...
2211
22222121
11212111
Applied Linear Algebra Instructor: Dr.V.R.K.REDDY
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23 December 2019 47
Theorem 1.2.1
◼ A homogeneous system of linear equations with more
unknowns than equations has infinitely many solutions.
◼ Remark
❑ This theorem applies only to homogeneous system!
❑ A nonhomogeneous system with more unknowns than equations
need not be consistent; however, if the system is consistent, it
will have infinitely many solutions.
❑ e.g., two parallel planes in 3-space
Applied Linear Algebra Instructor: Dr.V.R.K.REDDY
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12/23/2019 48
Chapter Contents
◼ Introduction to System of Linear Equations
◼ Gaussian Elimination
◼ Matrices and Matrix Operations
◼ Inverses; Rules of Matrix Arithmetic
◼ Elementary Matrices and a Method for Finding A-1
◼ Further Results on Systems of Equations and Invertibility
◼ Diagonal, Triangular, and Symmetric Matrices
Applied Linear Algebra Instructor: Dr.V.R.K.REDDY23 December 2019 48
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23 December 2019 49
1-3 Definition and Notation◼ A matrix is a rectangular array of numbers. The numbers
in the array are called the entries in the matrix
◼ A general mn matrix A is denoted as
◼ The entry that occurs in row i and column j of matrix Awill be denoted aij or Aij. If aij is real number, it is common to be referred as scalars
◼ The preceding matrix can be written as [aij]mn or [aij]
=
mnmm
n
n
aaa
aaa
aaa
A
...
...
...
21
22221
11211
Applied Linear Algebra Instructor: Dr.V.R.K.REDDY
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➢ A matrix with only one row, such as the second in
Example 1, is called a row vector (or a row matrix).
23 December 2019 Applied Linear Algebra Instructor: Dr.V.R.K.REDDY 50
➢ A matrix with only one column, such as the fourth in
that example, is called a column vector (or a column
matrix).
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23 December 2019 Applied Linear Algebra Instructor: Dr.V.R.K.REDDY 51
We will use capital letters to denote matrices and lowercase
letters to denote numerical quantities; thus we might write
When discussing matrices, it is common to refer to numerical
quantities as scalars. Unless stated otherwise, scalars will be
real numbers; complex scalars will be considered later in the
text/course.
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23 December 2019 Applied Linear Algebra Instructor: Dr.V.R.K.REDDY 52
When a compact notation is desired, the preceding matrix can be
written as
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23 December 2019 Applied Linear Algebra Instructor: Dr.V.R.K.REDDY 53
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23 December 2019 54
1-3 Definition
◼
Applied Linear Algebra Instructor: Dr.V.R.K.REDDY
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23 December 2019 Applied Linear Algebra Instructor: Dr.V.R.K.REDDY 56
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Matrix Multiplication
◼
23 December 2019 Applied Linear Algebra Instructor: Dr.V.R.K.REDDY 57
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23 December 2019 Applied Linear Algebra Instructor: Dr.V.R.K.REDDY 58
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23 December 2019 Applied Linear Algebra Instructor: Dr.V.R.K.REDDY 59
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23 December 2019 60
◼
Applied Linear Algebra Instructor: Dr.V.R.K.REDDY
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1-3 Example 5
◼ Multiplying matrices
=
062
421A
−=
2572
1310
3414
B
Applied Linear Algebra Instructor: Dr.V.R.K.REDDY23 December 2019 61
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◼
Applied Linear Algebra Instructor: Dr.V.R.K.REDDY23 December 2019 62
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23 December 2019 63
1-3 Partitioned Matrices
◼ A matrix can be partitioned into smaller matrices by inserting
horizontal and vertical rules between selected rows and columns
◼ For example, three possible partitions of a 34 matrix A:
❑ The partition of A into four
submatrices A11, A12, A21,
and A22
❑ The partition of A into its row
matrices r1, r2, and r3
❑ The partition of A into its
column matrices c1, c2, c3,
and c4 4321
34333231
24232221
14131211
3
2
1
34333231
24232221
14131211
2221
1211
34333231
24232221
14131211
cccc
r
r
r
=
=
=
=
=
=
aaaa
aaaa
aaaa
A
aaaa
aaaa
aaaa
A
AA
AA
aaaa
aaaa
aaaa
A
Applied Linear Algebra Instructor: Dr.V.R.K.REDDY
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23 December 2019 64
1-3 Multiplication by Columns and by Rows
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Multiplication by Columns and by Rows …Continue
23 December 2019 Applied Linear Algebra Instructor: Dr.V.R.K.REDDY 65
=
=
==
B
B
B
BAB
AAAAAB
mm
nn
a
a
a
a
a
a
bbbbbb
2
1
2
1
2121
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1-3 Example 7
◼ Multiplying matrices by rows and by columns
=
062
421A
−=
2572
1310
3414
B
Applied Linear Algebra Instructor: Dr.V.R.K.REDDY23 December 2019 66
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Matrix Products as Linear Combinations
◼
23 December 2019 Applied Linear Algebra Instructor: Dr.V.R.K.REDDY 67
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23 December 2019 Applied Linear Algebra Instructor: Dr.V.R.K.REDDY 68
1-3 Matrix Products as Linear Combinations
◼ Let
Then
=
=
nmnmm
n
n
x
x
x
aaa
aaa
aaa
A
2
1
21
22221
11211
and x
++
+
=
+++
+++
+++
=
mn
n
n
n
mmnmnmm
nn
nn
a
a
a
x
a
a
a
x
a
a
a
x
xaxaxa
xaxaxa
xaxaxa
A
2
1
2
22
12
2
1
21
11
1
2211
2222121
1212111
x
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23 December 2019 Applied Linear Algebra Instructor: Dr.V.R.K.REDDY 69
1-3 Example 8
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23 December 2019 Applied Linear Algebra Instructor: Dr.V.R.K.REDDY 70
1-3 Example 9
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23 December 2019 Applied Linear Algebra Instructor: Dr.V.R.K.REDDY 71
1-3 Matrix Form of a Linear System
◼ Consider any system of m linear equations in n unknowns:
◼ The matrix A is called the coefficient matrix of the system
◼ The augmented matrix of the system is given by
=
mmmnmm
n
n
b
b
b
x
x
x
aaa
aaa
aaa
2
1
2
1
21
22221
11211
nnmnmm
nn
nn
bxaxaxa
bxaxaxa
bxaxaxa
=+++
=+++
=+++
2211
22222121
11212111
=
+++
+++
+++
mnmnmm
nn
nn
b
b
b
xaxaxa
xaxaxa
xaxaxa
2
1
2211
2222121
1212111
bx =A
=
mmnmm
n
n
baaa
baaa
baaa
A
21
222221
111211
b
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1-3 Example 10
◼ A function using matrices
❑ Consider the following matrices
❑ The product y = Ax is
❑ The product y = Bx is
−=
10
01A
=
b
ax
−=
01
10B
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23 December 2019 Applied Linear Algebra Instructor: Dr.V.R.K.REDDY 73
1-3 Definitions
◼ If A is any mn matrix, then the transpose of A, denoted by AT, is defined to be the nm matrix that results from interchanging the rows and columns of A
❑ That is, the first column of AT is the first row of A, the second column of AT is the second row of A, and so forth
◼ If A is a square matrix, then the trace of A , denoted by tr(A), is defined to be the sum of the entries on the main diagonal of A. The trace of A is undefined if A is not a square matrix.
❑ For an nn matrix A = [aij], =
=n
i
iiaA1
)(tr
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1-3 Example 11 - Transpose:
◼
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1-3 Example 11 – Trace:
23 December 2019 Applied Linear Algebra Instructor: Dr.V.R.K.REDDY 75
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Exercise Problems
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12/23/2019 78
Chapter Contents
◼ Introduction to System of Linear Equations
◼ Gaussian Elimination
◼ Matrices and Matrix Operations
◼ Inverses; Rules of Matrix Arithmetic
◼ Elementary Matrices and a Method for Finding A-1
◼ Further Results on Systems of Equations and Invertibility
◼ Diagonal, Triangular, and Symmetric Matrices
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1-4 Properties of Matrix Operations
◼ For real numbers a and b ,we always have ab = ba, which
is called the commutative law for multiplication. For
matrices, however, AB and BA need not be equal.
◼ Equality can fail to hold for three reasons:
❑ The product AB is defined but BA is undefined.
❑ AB and BA are both defined but have different sizes.
❑ It is possible to have AB BA even if both AB and BA are
defined and have the same size.
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23 December 2019 Applied Linear Algebra Instructor: Dr.V.R.K.REDDY 80
Theorem 1.4.1 (Properties of Matrix Arithmetic)◼ Assuming that the sizes of the matrices are such that the indicated
operations can be performed, the following rules of matrix arithmetic are valid:
❑ A + B = B + A (commutative law for addition)
❑ A + (B + C) = (A + B) + C (associative law for addition)
❑ A(BC) = (AB)C (associative law for multiplication)
❑ A(B + C) = AB + AC (left distributive law)
❑ (B + C)A = BA + CA (right distributive law)
❑ A(B – C) = AB – AC, (B – C)A = BA – CA
❑ a(B + C) = aB + aC, a(B – C) = aB – aC
❑ (a+b)C = aC + bC, (a-b)C = aC – bC
❑ a(bC) = (ab)C, a(BC) = (aB)C = B(aC)
◼ Note: the cancellation law is not valid for matrix multiplication!
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23 December 2019 Applied Linear Algebra Instructor: Dr.V.R.K.REDDY 83
1-4 Zero Matrices
◼ A matrix, all of whose entries are zero, is called a zero
matrix
◼ A zero matrix will be denoted by 0
◼ If it is important to emphasize the size, we shall write
0mn for the mn zero matrix.
◼ In keeping with our convention of using boldface
symbols for matrices with one column, we will denote a
zero matrix with one column by 0
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1-4 Example 3
◼ The cancellation law does not hold
◼ AB=AC =
◼ AD =
=
20
10A
=
43
11B
=
43
52C
=
00
73D
86
43
00
00
23 December 2019 Applied Linear Algebra Instructor: Dr.V.R.K.REDDY 85
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Theorem 1.4.2 (Properties of Zero Matrices)
◼ THEOREM 1.4.2 Properties of Zero Matrices
If c is a scalar, and if the sizes of the matrices are such
that the operations can be perfomed, then:
23 December 2019 Applied Linear Algebra Instructor: Dr.V.R.K.REDDY 86
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23 December 2019
Applied Linear Algebra Instructor:
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1-4 Identity Matrices
◼ A square matrix with 1s on the main diagonal and 0s off the main diagonal is called an identity matrix and is denoted by I, or In for the nn identity matrix
◼ If A is an mn matrix, then AIn = A and ImA = A
❑ Example 4
◼ An identity matrix plays the same role in matrix arithmetic as the number 1 plays in the numerical relationships a·1 = 1·a = a
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Theorem 1.4.3
◼
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23 December 2019
Applied Linear Algebra Instructor:
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1-4 Invertible
◼ If A is a square matrix, and if a matrix B of the same size
can be found such that AB = BA = I, then A is said to be
invertible and B is called an inverse of A. If no such
matrix B can be found, then A is said to be singular.
◼ Remark:
❑ The inverse of A is denoted as A-1
❑ Not every (square) matrix has an inverse
❑ An inverse matrix has exactly one inverse
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1-4 Example 5 & 6
◼ Verify the inverse requirements
◼ A matrix with no inverse
is singular
−
−=
31
52A
=
21
53B
=
063
052
041
A
23 December 2019
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Dr.V.R.K.REDDY 91
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23 December 2019 Applied Linear Algebra Instructor: Dr.V.R.K.REDDY 92
1-4 Theorems◼ Theorem 1.4.4
❑ If B and C are both inverses of the matrix A, then B = C
◼ Theorem 1.4.5
❑ The matrix
is invertible if ad – bc 0, in which case the inverse is given by the
formula
=
dc
baA
−
−
−=−
ac
bd
bcadA
11
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Theorem 1.4.6
◼
=
31
21A
=
22
23B
=
89
67AB
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23 December 2019 Applied Linear Algebra Instructor: Dr.V.R.K.REDDY 94
1-4 Powers of a Matrix
◼ If A is a square matrix, then we define the nonnegative integer powers of A to be
◼ If A is invertible, then we define the negative integer powers to be
◼ Theorem 1.4.7 (Laws of Exponents)
❑ If A is a square matrix and r and s are integers, then ArAs = Ar+s, (Ar)s = Ars
)0(factors
0 == nAAAAIAn
n
)0()(factors
1111 == −−−−− nAAAAAn
nn
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Theorem 1.4.8 (Laws of Exponents)
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1-4 Example 8
◼ Powers of matrix
◼ A3
=
31
21A
−
−=−
11
231A
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23 December 2019
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1-4 Polynomial Expressions Involving Matrices
◼ If A is a square matrix, say mm , and if
p(x) = a0 + a1x + … + anxn
is any polynomial, then we define
p(A) = a0I + a1A + … + anAn
where I is the mm identity matrix.
◼ That is, p(A) is the mm matrix that results when A is
substituted for x in the above equation and a0 is replaced
by a0I
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23 December 2019
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1-4 Example 9 (Matrix Polynomial)
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23 December 2019 Applied Linear Algebra Instructor: Dr.V.R.K.REDDY 99
Theorems 1.4.9 (Properties of the Transpose)THEOREM 1.4.8
If the sizes of the matrices are such that the stated operations can
be performed, then:
Note:
The transpose of a product of any number of matrices is the
product of the transposes in the reverse order.
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Theorem 1.4.10 (Invertibility of a Transpose)
◼
−−=
12
35A
−
−=
13
25TA
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Exercise Problems
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23 December 2019 Applied Linear Algebra Instructor: Dr.V.R.K.REDDY 102
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23 December 2019 Applied Linear Algebra Instructor: Dr.V.R.K.REDDY 103
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12/23/2019 Elementary Linear Algorithm 104
Chapter Contents
◼ Introduction to System of Linear Equations
◼ Gaussian Elimination
◼ Matrices and Matrix Operations
◼ Inverses; Rules of Matrix Arithmetic
◼ Elementary Matrices and a Method for Finding A-1
◼ Further Results on Systems of Equations and Invertibility
◼ Diagonal, Triangular, and Symmetric Matrices
23 December 2019
Applied Linear Algebra Instructor:
Dr.V.R.K.REDDY 104
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23 December 2019 Applied Linear Algebra Instructor: Dr.V.R.K.REDDY 105
1-5 Elementary Row Operation
◼
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1-5 Elementary Matrix
◼
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23 December 2019 Applied Linear Algebra Instructor: Dr.V.R.K.REDDY 107
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23 December 2019
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1-5 Example 1
◼ Elementary Matrices and Row Operations
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23 December 2019
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1-5 Elementary Matrices and Row Operations
◼ Theorem 1.5.1
❑ Suppose that E is an mm elementary matrix produced by
applying a particular elementary row operation to Im, and
that A is an mn matrix. Then EA is the matrix that results
from applying that same elementary row operation to A
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23 December 2019
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1-5 Example 2 (Using Elementary Matrices)
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23 December 2019 Applied Linear Algebra Instructor: Dr.V.R.K.REDDY 111
1-5 Inverse Operations
◼
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1-5 Inverse Operations
◼ Examples
23 December 2019 Applied Linear Algebra Instructor: Dr.V.R.K.REDDY 112
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23 December 2019 Applied Linear Algebra Instructor: Dr.V.R.K.REDDY 113
Theorem 1.5.2
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23 December 2019
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Theorem 1.5.3(Equivalent Statements)
◼ If A is an nn matrix, then the following statements are
equivalent, that is, all true or all false
❑ A is invertible
❑ Ax = 0 has only the trivial solution
❑ The reduced row-echelon form of A is In
❑ A is expressible as a product of elementary matrices
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23 December 2019 Applied Linear Algebra Instructor: Dr.V.R.K.REDDY 115
1-5 A Method for Inverting Matrices
◼ To find the inverse of an invertible matrix A, we must
find a sequence of elementary row operations that reduces
A to the identity and then perform this same sequence of
operations on In to obtain A-1
◼ Remark
❑ Suppose we can find elementary matrices E1, E2, …, Ek such that
Ek … E2 E1 A = In
then
A-1 = Ek … E2 E1 In
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23 December 2019 Applied Linear Algebra Instructor: Dr.V.R.K.REDDY 116
◼ Find the inverse of
◼ Solution:
❑ To accomplish this we shall adjoin the identity matrix to the right
side of A, thereby producing a matrix of the form [A | I]
❑ We shall apply row operations to this matrix until the left side is
reduced to I; these operations will convert the right side to A-1, so
that the final matrix will have the form [I | A-1]
=
801
352
321
A
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23 December 2019 Applied Linear Algebra Instructor: Dr.V.R.K.REDDY 117
1-5 Example 4
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23 December 2019 Applied Linear Algebra Instructor: Dr.V.R.K.REDDY 118
1-5 Example 4 (continue)
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1-5 Example 5
◼ Consider the matrix
Apply the procedure of example 4 to find A-1
23 December 2019 Applied Linear Algebra Instructor: Dr.V.R.K.REDDY 119
−
−=
521
142
461
A
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1-5 Example 6
◼ According to example 4, A is an invertible matrix.
◼ has only trivial solution
23 December 2019 Applied Linear Algebra Instructor: Dr.V.R.K.REDDY 120
=
801
352
321
A
08
0 352
0 32
31
321
321
=++
=++
=++
xx
xxx
xxx
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12/23/2019 121
Chapter Contents
◼ Introduction to System of Linear Equations
◼ Gaussian Elimination
◼ Matrices and Matrix Operations
◼ Inverses; Rules of Matrix Arithmetic
◼ Elementary Matrices and a Method for Finding A-1
◼ Further Results on Systems of Equations and Invertibility
◼ Diagonal, Triangular, and Symmetric Matrices
23 December 2019 Applied Linear Algebra Instructor: Dr.V.R.K.REDDY 121
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23 December 2019 Applied Linear Algebra Instructor: Dr.V.R.K.REDDY 122
Theorems 1.6.1
◼ Every system of linear equations has either no
solutions, exactly one solution, or in finitely many
solutions.
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Theorem 1.6.2
◼ If A is an invertible nn matrix, then for each n1
matrix b, the system of equations Ax = b has exactly
one solution, namely, x = A-1b.
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1-6 Example 1
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1-6 Linear Systems
with a Common Coefficient Matrix
◼ To solve a sequence of linear systems, Ax = b1, Ax = b1, …,
Ax = bk, with common coefficient matrix A
◼ If A is invertible, then the solutions x1 = A-1b1, x2 = A-1b2 , …,
xk = A-1bk
◼ A more efficient method is to form the matrix [A|b1|b2|…|bk]
◼ By reducing it to reduced row-echelon form we can solve all k
systems at once by Gauss-Jordan elimination.
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23 December 2019
Applied Linear Algebra Instructor:
Dr.V.R.K.REDDY 127
Theorems 1.6.3
◼ Let A be a square matrix
❑ If B is a square matrix satisfying BA = I, then B = A-1
❑ If B is a square matrix satisfying AB = I, then B = A-1
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Theorem 1.6.4 (Equivalent Statements)
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Theorem 1.6.5
◼
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Exercise Problem
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136
Chapter Contents
◼ Introduction to System of Linear Equations
◼ Gaussian Elimination
◼ Matrices and Matrix Operations
◼ Inverses; Rules of Matrix Arithmetic
◼ Elementary Matrices and a Method for Finding A-1
◼ Further Results on Systems of Equations and Invertibility
◼ Diagonal, Triangular, and Symmetric Matrices
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1-7 Diagonal Matrix
◼ A square matrix A is mn with m = n; the (i,j)-entries
for 1 i m form the main diagonal of A
◼ A diagonal matrix is a square matrix all of whose
entries not on the main diagonal equal zero. By
diag(d1, …, dm) is meant the mm diagonal matrix
whose (i,i)-entry equals di for 1 i m
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1-7 Properties of Diagonal Matrices
◼ A general nn diagonal matrix
D can be written as
◼ A diagonal matrix is invertible
if and only if all of its diagonal
entries are nonzero
◼ Powers of diagonal matrices
are easy to compute
=
nd
d
d
D
00
00
00
2
1
=−
nd
d
d
D
/100
0/10
00/1
2
1
1
=
k
n
k
k
k
d
d
d
D
00
00
00
2
1
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1-7 Properties of Diagonal Matrices
◼ Matrix products that involve diagonal factors are
especially easy to compute
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1-7 Triangular Matrices◼
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Note: A square matrix in which all the entries above the main
diagonal are zero is called lower triangular, and a square matrix in
which all the entries below the main diagonal are zero is called upper
triangular. A matrix that is either upper triangular or lower triangular
is called triangular.
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1-7 Example 2 (Triangular Matrices)
◼ The diagonal matrix
❑ both upper triangular and lower triangular
◼ A square matrix in row-echelon form is upper triangular
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Theorem 1.7.1The following theorem lists some of the basic properties of
triangular matrices. ◼ The transpose of a lower triangular matrix is upper triangular,
and the transpose of an upper triangular matrix is lower triangular
◼ The product of lower triangular matrices is lower triangular, and the product of upper triangular matrices is upper triangular
◼ A triangular matrix is invertible if and only if its diagonal entries are all nonzero
◼ The inverse of an invertible lower triangular matrix is lower triangular, and the inverse of an invertible upper triangular matrix is upper triangular
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1-7 Example 3
◼ Consider the upper triangular matrices
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−
=
500
420
131
A
−
−
=
100
100
223
B
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1-7 Symmetric Matrices
◼
−
705
034
541
−
−
73
37
4
3
2
1
000
000
000
000
d
d
d
d
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Theorem 1.7.2
◼
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◼
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1-7 Example 6
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D E F IN I T ION: Pivot Positions
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