Download Chapter 1 Matrices and Systems of Equations

Chapter 1---Section 1 Systems of Linear Equations
Chapter 1 Matrices and Systems of Equations
Linear systems arise in applications to such areas as engineering,
physics, electronics, business, economics, sociology（社会学）, ecology
（生态学）, demography(人口统计学), and genetics（遗传学）, etc.
§1. Systems of Linear Equations
New words and phrases in this section:
Linear equation 线性方程
Linear system，System of linear equations 线性方程组
Unknown 未知量
Consistent 相容的
Consistence 相容性
Inconsistent 不相容的
Inconsistence 不相容性
Solution 解
Solution set 解集
Equivalent 等价的
Equivalence 等价性
Equivalent system 等价方程组
Strict triangular system 严格上三角方程组
Strict triangular form 严格上三角形式
Back Substitution 回代法
Matrix 矩阵
Coefficient matrix 系数矩阵
Augmented matrix 增广矩阵
Pivot element 主元
Pivotal row 主行
Echelon form 阶梯形
1.1 Definitions
A linear equation (线性方程) in n unknowns（未知量） is
a1 x1  a2 x2  ...  an xn  b
Chapter 1---Section 1 Systems of Linear Equations
A linear system of m equations in n unknowns is
 a11 x1  a12 x2  ...  a1n xn  b1

 a21 x1  a22 x2  ...  a2 n xn  b2

......
 am1 x1  am 2 x2  ...  amn xn  bm
This is called a mxn(read as m by n) system.
A solution to an mxn system is an ordered n-tuple of numbers （n 元
数组） ( x1 , x2 ,..., xn ) that satisfies all the equations.
A system is said to be inconsistent（不相容的） if the system has no
solutions.
A system is said to be consistent（相容的）if the system has at least
one solution.
The set of all solutions to a linear system is called the solution set
（解集）of the linear system.
1.2 Geometric Interpretations of 2x2 Systems
a11 x1  a12 x2  b1

a21 x1  a22 x2  b2
Each equation can be represented graphically as a line in the plane.
The ordered pair
( x1 , x2 )
will be a solution if and only if it lies on both
lines.
In the plane, the possible relative positions are
(1) two lines intersect at exactly a point; (The solution set has exactly
one element)
Chapter 1---Section 1 Systems of Linear Equations
(2) two lines are parallel; (The solution set is empty)
(3) two lines coincide. (The solution set has infinitely many
elements)
The situation is the same for mxn systems. An mxn system may not
be consistent. If it is consistent, it must either have exactly one solution or
infinitely many solutions. These are only possibilities.
Of more immediate concerns is the problem of finding all solutions to
a given system.
1.3 Equivalent systems
Two systems of equations involving the same variables are said to be
equivalent （等价的，同解的）if they have the same solution set.
To find the solution set of a system, we usually use operations to
reduce the original system to a simpler equivalent system.
It is clear that the following three operations do not change the
solution set of a system.
(1)
Interchange the order in which two equations of a system are
written;
(2)
Multiply through one equation of a system by a nonzero real
number;
(3)
Add a multiple of one equation to another equation. (subtract
Chapter 1---Section 1 Systems of Linear Equations
a multiple of one equation from another one)
Remark: The three operations above are very important in dealing
with linear systems. They coincide with the three row operations of
matrices. Ask a student about the proof.
1.4 n x n systems
If an nxn system has exactly one solution, then operation 1 and 3
can be used to obtain an equivalent “strictly triangular system”
A system is said to be in strict triangular form (严格三角形) if in the
k-th equation the coefficients of the first k-1 variables are all zero and the
coefficient of
xk is
nonzero. (k=1, 2, …,n)
An example of a system in strict triangular form:
3 x1  3 x2  x3  1

x2  x3  2


2 x3  4

Any nxn strictly triangular system can be solved by back substitution
(回代法).
(Note: A phrase: “substitute 3 for x” == “replace x by 3”)
In general, given a system of linear equations in n unknowns, we
will use operation I and III to try to obtain an equivalent system that is
strictly triangular.
We can associate with a linear system an mxn array of numbers
whose entries are coefficient of the
xi ’s.
we will refer to this array as the
Chapter 1---Section 1 Systems of Linear Equations
coefficient matrix (系数矩阵) of the system.
 a11

 a21
 ...

 am1
a12
a22
...
am 2
... a1n 

... a2 n 
... ... 

... amn 
A matrix （矩阵） is a rectangular array of numbers
If we attach to the coefficient matrix an additional column whose
entries are the numbers on the right-hand side of the system, we obtain
the new matrix
 a11
 a21

a
 m1
a12
a22
a1n
a2 n
am 2
amn
b1 
b2 

bs 
We refer to this new matrix as the augmented matrix（增广矩阵） of
a linear system.
The system can be solved by performing operations on the
augmented matrix. xi ’s are placeholders that can be omitted until the end
of computation.
Corresponding to the three operations used to obtain equivalent
systems, the following row operation may be applied to the augmented
matrix.
Chapter 1---Section 1 Systems of Linear Equations
1.5 Elementary row operations
There are three elementary row operations:
(1) Interchange two rows;
(2) Multiply a row by a nonzero number;
(3) Replace a row by its sum with a multiple of another row.
Remark: The importance of these three operations is that they do not
change the solution set of a linear system and may reduce a linear system
to a simpler form.
An example is given here to illustrate how to perform row operations
on a matrix.
★ Example:
The procedure for applying the three elementary row operations:
Step 1: Choose a pivot element (主元)(nonzero) from among the
entries in the first column. The row containing the pivot
number is called a pivotal row（主行）. We interchange the
rows (if necessary) so that the pivotal row is the new first
row.
Multiples of the pivotal row are then subtracted form each
of the remaining n-1 rows so as to obtain 0’s in the first
entries of rows 2 through n.
Step2: Choose a pivot element from the nonzero entries in column
2, rows 2 through n of the matrix. The row containing the
Chapter 1---Section 1 Systems of Linear Equations
pivot element is then interchanged with the second row ( if
necessary) of the matrix and is used as the new pivotal row.
Multiples of the pivotal row are then subtracted form each
of the remaining n-2 rows so as to eliminate all entries below
the pivot element in the second column.
Step 3: The same procedure is repeated for columns 3 through n-1.
Note that at the second step, row 1 and column 1 remain unchanged,
at the third step, the first two rows and first two columns remain
unchanged, and so on.
At each step, the overall dimensions of the system are effectively
reduced by 1. (The number of equations and the number of unknowns all
decrease by 1.)
If the elimination process can be carried out as described, we will
arrive at an equivalent strictly triangular system after n-1 steps.
However, the procedure will break down if all possible choices for
a pivot element are all zero. When this happens, the alternative is to
reduce the system to certain special echelon form（梯形矩阵）.
Assignment
Students should be able to do all problems.
Hand-in problems are: # 7--#11
Chapter 1---Section 2
Row Echelon form
§2. Row Echelon Form
New words and phrases:
Row echelon form 行阶梯形
Reduced echelon form 简化阶梯形
Lead variable 首变量
Free variable 自由变量
Gaussian elimination 高斯消元
Gaussian-Jordan reduction. 高斯-若当消元
Overdetermined system 超定方程组
Underdetermined system
Homogeneous system 齐次方程组
Trivial solution 平凡解
2.1 Examples and Definition
In this section, we discuss how to use elementary row operations to
solve mxn systems.
Use an example to illustrate the idea.
★ Example: Example 1 on page 13. Consider a system represented
by the augmented matrix
1 1

 1 1
 2 2

0 0
1 1

1 1 1
0 0 1
0 0 3
1 1 3
2 2 4
1  1
 
1  0
1 0
 
1  0
1   0
1 1 1 1
0 1 1 2
0 2 2 5
0 1 1 3
0 1 1 3
1

0
3

1
0 
………..(The details will given in class)
We see that at this stage the reduction to strict triangular form breaks
down. Since our goal is to simplify the system as much as possible, we
move over to the third column. From the example above, we see that
the coefficient matrix that we end up with is not in strict triangular form,
Chapter 1---Section 2
Row Echelon form
it is in staircase or echelon form(梯形矩阵).
1

0
0

0
0

1 1 1 1
0 1 1 2
0 0 0 1
0 0 0 0
0 0 0 0
1

0
3

4 
3 
The equations represented by the last two rows are:
 x1  x2  x3  x4  x5  1

x3  x4  2 x5 =0

2 x5 =3


0=  4

0  3

Since there are no 5-tuples that could possibly satisfy these equations, the
system is inconsistent.
Change the system above to a consistent system.
1 1

 1 1
 2 2

0 0
1 1

1 1 1
0 0 1
0 0 3
1 1 3
2 2 4
1  1
 
1  0
1 0
 
3  0
4   0
1 1 1 1 1

0 1 1 2 0
0 0 0 1 3

0 0 0 0 0
0 0 0 0 0 
The last two equations of the reduced system will be satisfied for
any 5-tuple. Thus the solution set will be the set of all 5-tuples satisfying
the first 3 equations.
The variables corresponding to the first nonzero element in each row
of the augment matrix will be referred to as lead variable.（首变量） The
remaining variables corresponding to the columns skipped in the
reduction process will be referred to as free variables（自由变量）.
Chapter 1---Section 2
Row Echelon form
If we transfer the free variables over to the right-hand side in the
above system, then we obtain the system:
 x1  x3  x5  1  x2  x4

x3  2 x5   x4


x5  3

which is strictly triangular in the unknown x1 x3 x5 . Thus for each pair
of values assigned to x2 and x4 , there will be a unique solution.
★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 number 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.
★Definition: The process of using row operations I, II and III to
transform a linear system into one whose augmented matrix is in row
echelon form is called Gaussian elimination（高斯消元法）.
Note that row operation II is necessary in order to scale the rows so
that the lead coefficients are all 1.
Chapter 1---Section 2
Row Echelon form
It is clear that if the row echelon form of the augmented matrix
contains a row of the form  0 0
0 | 1 , the system is inconsistent.
Otherwise, the system will be consistent.
If the system is consistent and the nonzero rows of the row echelon
form of the matrix form a strictly triangular system (the number of
nonzero rows<the number of unknowns), the system will have a unique
solution. If the number of nonzero rows<the number of unknowns, then
the system has infinitely many solutions. (There must be at least one free
variable. We can assign the free variables arbitrary values and solve for
the lead variables.)
2.2 Overdetermined Systems
A linear system is said to be overdetermined if there are more
equations than unknowns.
2.3 Underdetermined Systems
A system of m linear equations in n unknowns is said to be
underdetermined if there are fewer equations than unknowns (m<n). It is
impossible for an underdetermined system to have only one solution.
Chapter 1---Section 2
Row Echelon form
In the case where the row echelon form of a consistent system has
free variables, it is convenient to continue the elimination process until all
the entries above each lead 1 have been eliminated. The resulting reduced
matrix is said to be in reduced row echelon form. For instance,
1

0
0

0
0

1 1 1 1 1 1
 
0 1 1 2 0 0
0 0 0 1 3 0
 
0 0 0 0 0 0
0 0 0 0 0   0
1 0 0 0
0 1 1 0
0 0 0 1
0 0 0 0
0 0 0 0
4

6 
3

0
0 
Put the free variables on the right-hand side, it follows that
x1  4  x2
x3  6  x4
x5  3
Thus for any real numbers  and  , the 5-tuple
4 
 6    3
is a solution.
Thus all ordered 5-tuple of the form  4    6    3 are
solutions to the system.
2.4 Reduced Row Echelon Form
★Definition: A matrix is said to be in reduced row echelon form
if :
(i)
the matrix is in row echelon form.
Chapter 1---Section 2
(ii)
Row Echelon form
The first nonzero entry in each row is the only nonzero entry
in its column.
The process of using elementary row operations to transform a
matrix into reduced echelon form is called Gaussian-Jordan
reduction.
The procedure for solving a linear system:
(i) Write down the augmented matrix associated to the system;
(ii) Perform elementary row operations to reduce the augmented
matrix into a row echelon form;
(iii)
If the system if consistent, reduce the row echelon form into a
reduced row echelon form.
(iv)
Write the solution in an n-tuple form
Remark: Make sure that the students know the difference between
the row echelon form and the reduced echelon form.
Example 6 on page 18: Use Gauss-Jordan reduction to solve the
system:
 x1  x2  x3  3x4  0

3x1  x2  x3  x4  0
2 x  x  2 x  x  0
3
4
 1 2
The details of the solution will be given in class.
2.5 Homogeneous Systems
A system of linear equations is said to be homogeneous if the
Chapter 1---Section 2
Row Echelon form
constants on the right-hand side are all zero.
Homogeneous systems are always consistent since it has a trivial
solution. If a homogeneous system has a unique solution, it must be the
trivial solution.
In the case that m<n (an underdetermined system), there will always
free variables and, consequently, additional nontrivial solution.
Theorem 1.2.1 An mxn homogeneous system of linear equations
has a nontrivial solution if m<n.
Proof A homogeneous system is always consistent. The row echelon
form of the augmented matrix can have at most m nonzero rows. Thus
there are at most m lead variables. There must be some free variable. The
free variables can be assigned arbitrary values. For each assignment of
values to the free variables, there is a solution to the system.
Assignment
Students should be able to do all problems except 17, 18, 20.
Hand-in problems are 9, 10, 16,
Select one problem from 14 and 19.
Chapter 1---Section3
Matrix Algebra
§3. Matrix Algebra
New words and phrases:
Algebra 代数
Scalar 数量，标量
Scalar multiplication 数乘
Real number 实数
Complex number 复数
Vector 向量
Row vector 行向量
Column vector 列向量
Euclidean n-space n 维欧氏空间
Linear combination 线性组合
Zero matrix 零矩阵
Identity matrix 单位矩阵
Diagonal matrix 对角矩阵
Triangular matrix 三角矩阵
Upper triangular matrix 上三角矩阵
Lower triangular matrix 下三角矩阵
Transpose of a matrix 矩阵的转置
(Multiplicative ) Inverse of a matrix 矩阵的逆
Singular matrix 奇异矩阵
Singularity 奇异性
Nonsingular matrix 非奇异矩阵
Nonsingularity 非奇异性
The term scalar (标量，数量) is referred to as a real number (实数) or
a complex number（复数）.
Matrix notation
An mxn matrix, a rectangular array of mn numbers.
 a11

 a21
 ...

 am1
a12
a22
...
am 2
... a1n 

... a2 n 
... ... 

... amn 
A  (aij )
Chapter 1---Section3
Matrix Algebra
3.1 Vectors
Matrices that have only one row or one column are of special interest
since they are used to represent solutions to linear systems.
We will refer to an ordered n-tuple of real numbers as a vector（向
量）.
If an n-tuple is represented in terms of a 1xn matrix, then we will
refer to it as a row vector. Alternatively, if the n-tuple is represented by
an nx1 matrix, then we will refer to it as a column vector.
In this course, we represent a vector as a column vector.
The set of all nx1 matrices of real number is called Euclidean n-space
(n 维欧氏空间) and is usually denoted by Rn .
Given a mxn matrix A, it is often necessary to refer to a particular row
or column. The matrix A can be represented in terms of either its column
vectors or its row vectors.
A  (a1 , a 2 ,
 a(1,:) 


a(2,:) 

, a n ) or A 




 a(m,:) 
Chapter 1---Section3
Matrix Algebra
3.2 Equality
For two matrices to be equal, they must have the same dimensions and
their corresponding entries must agree
★Definition: Two mxn matrices A and B are said to be equal if
aij  bij for each ordered pair (i, j)
3.3 Scalar Multiplication
If A is a matrix,  is a scalar, then  A is the mxn matrix formed by
multiplying each of the entries of A by  .
★Definition: If A is an mxn matrix,  is a scalar, then  A is the
mxn matrix whose (i, j) is  aij for each ordered pair (i, j) .
3.4 Matrix Addition
Two matrices with the same dimensions can be added by adding
their corresponding entries.
★Definition: If A and B are both mxn matrices, then the sum A+B
is the mxn matrix whose (i,j) entry is aij  bij for each ordered pair (i, j).
An mxn zero matrix (零矩阵) is a matrix whose entries are all zero. It
acts as an additive identity on the set of all mxn matrices.
A+O=O+A=A
Chapter 1---Section3
Matrix Algebra
The additive of A is (-1)A since A+(-1)A=O=(-1)A+A.
A-B=A+(-1)B
-A=(-1)A
3.5 Matrix Multiplication and Linear Systems
3.5.1 Motivations
Represent a linear system as a matrix equation
We have yet to defined the most important operation, the
multiplications of two matrices.
A 1x1 system can be written
ax  b
A scalar can be treated as a 1x1 matrix. Our goal is to generalize the
equation above so that we can represent an mxn system by a single
equation.
AX  B
Case 1: 1xn systems a1 x1  a2 x2  ...  an xn  b
If we set A   a1 a2
an 
 x1 
 
x
and X   2  , and define
 
 
 xn 
AX  a1 x1  a2 x2  ...  an xn
Then the equation can be written as AX  b 。
Case 2： mxn systems
Chapter 1---Section3
Matrix Algebra
Consider an mxn system, and let A be the coefficient matrix, X
the vector of unknowns, and B the vector of constants on the
right-hand side, and define the product AX by ….., then the linear
system is equivalent to the matrix equation AX=B.
3.5.2 Matrix Multiplication
More generally, it is possible to multiply a matrix A times a matrix B
if the number of columns of A equals the number of rows of B.
AB  ( Ab1 , Ab2 ,
, Ab n )
★Definition If A   aij  is an mxn matrix and B   bij  is an nxr
matrix, then the product AB  C   cij  is the mxr matrix whose entries
are defined by
n
cij   aik bkj
k 1
An alternative way to represent the linear system as a matrix equation
is to express the product AX as a sum of vectors.
★ Definition a1 , a 2 , , a n are vectors in R m , and c1 , c2 , , cn are
scalars, then a sum of the form c1a1 +c2a 2   cna n is said to a linear
combination of the vectors a1 , a 2 , , a n .
It follows that the product AX is a linear combination of the columns
vector of A.
Chapter 1---Section3
Matrix Algebra
AX  x1a1 +x2a 2 
x1a1 +x2a 2 
 xn a n
 xn a n  B provides a nice way of characterizing whether a
linear system of equations is consistent.
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.
Example 6 on page 37
3.6 Notational Rules
If an expression involves both multiplication and addition and there are
no parentheses to indicate the order of operations, multiplications are
carried out before additions. This is true for both scalar and matrix
multiplication.
3.7 Algebraic Rules
Theorem 1.3.2
Each of the following statements is valid for any
scalars  and  and for any matrices A, B, and C for which the
indicated operations are defined.
Properties 1---9
Chapter 1---Section3
Matrix Algebra
1. A  B  B  A (Commutative law of addition)
2. ( A  B)  C  A  ( B  C ) (Associative law of addition)
3. ( AB)C  A( BC ) (Associative law of matrix multiplication)
4. A( B  C )  AB  AC (Left distributive law)
5. ( A  B)C  AC  BC (Right distributive law)
6. ( ) A   (  A)
7.  ( AB)  ( A) B  A( B)
8. (   ) A   A   A
9.  ( A  B)   A   B (Distributive law)
We prove the associative law of matrix multiplication.
The details of the proof will be given in class.
Warning: In general, Matrix multiplication is not commutative.
Notation:
Ak  AA
A.
k times
3.8 Diagonal and Triangular Matrices
An nxn matrix A is said to be upper triangular（上三角的）if aij  0
for i>j and lower triangular （下三角的） if aij  0 for i<j. Also, A is
said to triangular if it is either upper triangular of lower triangular.
1 8

0 3
A 4x4 upper triangular matrix 
0 0

0 0
1 2

1 0
0 6

0 5
Chapter 1---Section3
Matrix Algebra
1 0

8 3
A 4x4 lower triangular matrix 
1 1

2 0
0 0

0 0
0 0

6 5
An nxn matrix is said to be diagonal（对角的） if aij  0 whenever
i  j.
1 0 0
A 3x3 diagonal matrix  0 2 0 
0 0 5


3.9 The Identity Matrix
Just as the number 1 acts an identity for the multiplications of real
numbers, there is a special matrix I that acts as an identity for matrix
multiplication, that is
IA=AI=A for any nxn matrix A.
★Definition The nxn identity matrix（单位矩阵） if the matrix
I   ij  , where
1 if i  j
0 if i  j
 ij  
1 0 0
A 3x3 identity matrix  0 1 0 
0 0 1


Chapter 1---Section3
Matrix Algebra
3.10 Matrix Inversion
A real number a is said to have a multiplicative inverse (乘法逆) if
there exists a number b such that ab=1. Any nonzero number a has a
multiplicative inverse b=1/a. We generalize the concept of multiplicative
inverse to matrices.
★Definition: An nxn matrix A is said to be nonsingular or invertible
if there exists a matrix B such that AB=BA=I. The matrix B is said to be a
multiplicative inverse (or simply inverse) of A.
The multiplicative inverse of a matrix A is unique. We denote it by
A1 .
★Definition An nxn matrix A is said to be singular if it does not have
a multiplicative inverse.
0 1
Question: Does the matrix A  
 have a multiplicative inverse?
0 0
Theorem 1.3.3
If A and B are nonsingular nxn matrices, then AB is
also nonsingular and  AB   B 1 A1 .
1
The proof will be given in class.
3.11 The Transpose of a Matrix
Given an mxn matrix A, it is often useful to form a new nxn matrix
whose columns are the rows of A.
Chapter 1---Section3
Matrix Algebra
★Definition The transpose （转置） of an nxm matrix A is the nxm
matrix B defined by
bij  a ji
for j=1, 2, …, n and i=1, 2, …, m. The transpose of is denoted by AT
Algebraic Rules for Transpose
1.
A 
T T
A
2. ( A)T   AT
3. ( A  B)T  AT  BT
4. ( AB)T  BT AT
We prove the 4th property in class.
★Definition An nxn matrix A is said to be symmetric（对称的） if
AT  A .
 1 1 0 3 


1 3 2 10 

A 5x5 symmetric matrix:
 0 2 9 5 


 3 10 5 2 
Assignment
Not required problems #30, #31
Hand-in problems:
11. 12. 13. 15. 16. 17. 22. 24. 27.
Chapter 1---Section 4
Elementary Matrices
§4. Elementary matrices
New words and phrases:
Elementary matrix 初等矩阵
Premultiply 左乘
Postmultiply 右乘
4.1 Objectives
In section 2, we learned the process of solving a linear system in terms
of row operations. In this section, we view this process in terms of matrix
multiplications.
Given a linear system AX=B, we can multiply both sides by a sequence
of special matrices to obtain an equivalent system in row echelon form.
The special matrices we will use are called elementary matrices.
We will use them to see how to compute the inverse of a nonsingular
matrix and also to obtain an important matrix factorization.
We begin by considering the effects of multiplying both sides of a
linear system by a nonsingular matrix.
4.2 Equivalent Systems
If M is a nonsingular matrix, then the following two systems are
equivalent.
(1)
AX=B
(2) MAX=MB
In other words, given an mxn linear system AX=B, we can obtain an
Chapter 1---Section 4
Elementary Matrices
equivalent system by multiplying both sides of the equation by a
nonsingular mxm matrix M.
The system represented by  A B  is equivalent to the system
represented by  MA MB  if M is nonsingular.
To obtain an equivalent system that is easier to solve, we can apply a
sequence of nonsingular matrices E1 , E2 , Ek to both sides of the
equation AX=B to obtain a simpler system UX=C.
E1 AX  E1B
E2 E1 AX  E2 E1B
Ek
E2 E1 AX  Ek
E2 E1B .
The question is: How to choose those matrices E1 , E2 , Ek so that we
can obtain a simpler system?
Recall that we can use row operations to get a new system that is
equivalent to the old one.
How the row operations and matrix multiplications are related?
4.3 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.
Chapter 1---Section 4
Elementary Matrices
So we can obtain three types of elementary matrices.
Type I. An elementary matrix of type I is a matrix obtained by
interchanging two rows of I.
Type II. An elementary matrix of type II is a matrix obtained by
multiplying a row of I by a nonzero number.
Type III. An elementary matrix of type III is a matrix obtained from I
by adding a multiple of one row to another row.
Use elementary matrices to multiply a matrix on the left to observe
the effects.
In general, suppose that E is an nxn elementary matrix. We can think
of E as being obtained from I by either a row operation or a column
operation. If A is an nxr matrix, premultiplying(左乘) A by E has the
effect of performing that same row operation on A. If B is an mxn matrix,
postmultiplying（右乘） B by E is equivalent to performing that same
column operation on B.
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.
Proof Construct the inverse of E for each type, and give the specific
representation of the inverse.
Chapter 1---Section 4
Important Fact:
Elementary Matrices
Premultiplying a matrix A or postmultiplying a
matrix A by an elementary matrix is equivalent to performing a row
operation or a column operation.
★Definition A matrix B is row equivalent to A if there exists a finite
sequence of E1 , E2 , Ek of elementary matrices such that
Ek
E2 E1B  A
(An equivalent definition for row equivalence: A is row equivalent to B
if there is a nonsingular matrix M such that A=MB.)
( In other words, B is row equivalent to A if B can be obtained from A
by a finite number of row operations. In particular, two augmented
matrices (A|b) and (B|c) are equivalent if and if AX=b and Bx=c are
equivalent systems.
Two properties of row equivalent matrices: (of course A is equivalent
to itself, reflexivity 自反性)
I.
If A is equivalent to B, the B is equivalent to A (symmetry 对称
性)
II.
If A is equivalent to B, and B is equivalent to C, then A is
equivalent to C. (Transitivity 传递性)
Theorem 1.4.2 (Equivalent Conditions for Nonsingularity) Let A be an
nxn matrix. The following are equivalent:
(a) A is nonsingular.
Chapter 1---Section 4
Elementary Matrices
(b) AX=0 has only the trivial solution 0.
(c) A is row equivalent to I.
Proof (a) implies (b)
Multiply both sides of the equation by the inverse of A.(two
systems are equivalent)
(b) implies (c)
Use row operations to transform the system into the form UX=0,
where U is in row echelon form. If one of the diagonal elements of U
were 0, the last row of U would consist entirely of 0’s. But then AX=0
would be equivalent to a system with at least free variable and hence
would have a nontrivial solution. Thus U must be a upper triangular
matrix with diagonal elements all equal to 1. It follows then that I is the
reduced row echelon form of A and hence A is now equivalent to I.
(d)
implies (a)
If A is row equivalent to the identity matrix I, then A can be written as
a product of a finite sequence of elementary matrices. All elementary
matrices are invertible, so the product is also invertible. Hence, A is
nonsingular.
Corollary 1.4.3 An nxn system AX=B has a unique solution if and only if
A is nonsingular.
Chapter 1---Section 4
Elementary Matrices
Proof If A is nonsingular, then by premultiplying both sides of the
equation by the inverse of A, and conclude that the solution must be equal
to A1B .
Conversely, if AX=B has a unique solution X, then we claim A cannot
be singular. Indeed, if A were singular, then the equation AX=0 would
have a nontrivial solution Z (theorem 1.4.2). But this imply that Y=X+Z
is a second solution to AX=B. Therefore, A must be nonsingular.
Theorem 1.4.2 actually tells us a way to find the inverse of A if A is
nonsingular.
A is nonsingular if and only if A is row equivalent to the identity
matrix I, and hence there are elementary matrices E1 , E2 , Ek such that
Ek
Thus Ek
E2 E1 A  I
E2 E1 I  A1 .
This implies that the same series of elementary row operations that
transforms a nonsingular matrix into I will transform I into A1 . Thus, if
we augment A by I and perform the elementary row operations that
transform A into I on the augmented matrix, then I will be transformed
into A1 .
Example Compute A1 if
 1 4 3


A   1 2 0 
 2 2 3


Chapter 1---Section 4
1 4 3

 1 2 0
2 2 3



1 0 0 1 0 0

0 1 0    0 1 0
0 0 1   0 0 1



Elementary Matrices
1
2
1
4
1
6

1
2
1

4
1
2

1 
2 
1
 
4

1 
6 
If we want to solve the system represented by the augmented matrix
1 4 3

 1 2 0
2 2 3

12 

12 
8 
Then what we need to do is to do a matrix multiplication.
Assignment for 1.4
Hand-in problems: 5, 7, 13, 15, 18, 22, 26, 27.
Chapter 1---Section 5 Partitioned Matrices
§5 Partitioned Matrices
New words and phrases
Partition 分块，分割
Submatrix 子矩阵
Block multiplication 分块乘法
Often it is useful to think of a matrix as being composed of a number
of submatrices（子矩阵）. A matrix C can be partitioned into smaller
matrices by drawing horizontal lines between the rows and vertical lines
between the columns. For example, we can partition（分块）a matrix into
a row submatrices or column submatrices.
If B is partitioned into column submatrices, B  (b1 ,b2 , ,bn ) then
AB  (Ab1 ,Ab2 ,
,Abn )
 a(1,:) 


a(2,:) 

If A is partitioned into row submatrices, A 
， then




 a(m,:) 
 a(1,:)B 


a(2,:)B 

AB 




 a(m,:)B 
Next we consider how to compute the product AB in terms of more
general partitions of A and B.
Block Multiplication (分块乘法)
Let A be an mxn matrix and B an mxr matrix. It is often useful to
partition A and B and express the product in terms of the submatrices of A
Chapter 1---Section 5 Partitioned Matrices
and B. Consider the following four cases.
Recall two basic cases.
And then present the following four cases.
Case 1 Use one vertical line to partition B into two submatrices.
B   B1 B2  , then AB   AB1
AB2  (this can be proven
by using the basic cases)
Case 2 Use a horizontal line to partition A into two submatrices
 A1 
 A1B 
A    , then AB  

 A2 
 A2 B 
(this can be proven by using the basic cases)
Case 3 Use one vertical line to partition A into two submatrices and
use one horizontal line to partition B into two submatrices
A   A1
B 
A2  , B   1  then AB= A1B1  A2 B2
 B2 
Notice that this can not be proved by using the basic cases or case 2.
This can be proven by dividing the sum into two groups
n
s
n
k 1
k 1
k s
cij   aik bkj   aik bkj   aik bkj
Case 4 Use one horizontal line and one vertical line to partition both
A and B into four submatrices.
A
A 
B
B 
If A   11 12  B   11 12  , then
 A21 A22 
 B21 B22 
 A11B11  A12 B21
 A21B11  A22 B21
AB= AB  
A11B12  A12 B22 

A21B12  A22 B22 
This can be proven by using case 3.
Chapter 1---Section 5 Partitioned Matrices
Example 2 on page 77
 A11

O
Let A be an nxn matrix of the form
O

A22 
where A11 is a kxk matrix (k<n). Show that A is nonsingular if and
only if A11 and A22 are nonsingular.
Proof: Sufficiency
If A11 and A22 are nonsingular….
Necessity
If A is nonsingular, then let B  A1 and partition B in the same
manner as A.
 B11

 B21
B12   A11

B22   O
O   A11
 =
A22   O
O   B11

A22   B21
B12   I k
=
B22   O
O 
 =I
I nk 
And show that A11 and A22 are nonsingular.
Case 5 The general case
In general, if the blocks have proper dimensions, the block
multiplication can be carried out in the same manner as ordinary matrix
multiplication. If
 A11

A
A
 s1
A1t 


Ast 
 B1 1

B
B
 t1
B r 1

 Then
B t r
 C11

AB  
C
 s1
t
C1r 


Csr 
Ci j   A i kB . k The
multiplication can be carried out if the number of
j
k 1
columns of Aik equals the number of rows of Bkj for each k.
Chapter 1---Section 5 Partitioned Matrices
Assignment
“Hand in” problems: 2, 7, 9, 11, 14, 15, 16, 20,
Chapter 1---Section 5 Partitioned Matrices