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Chapter 1

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Chapter 1

Matrices and Systems of Equations

- Systems of Linear Equations

Where the aij’sand bi’s are all real numbers, xi’s are variables . We will refer to systems of the form (1) as m×n linear systems.

Definition

Inconsistent : A linear system has no solution.

Consistent : A linear system has at least one solution.

Example

(ⅰ) x1 + x2 = 2

x1 − x2 = 2

(ⅱ) x1 + x2 = 2

x1+ x2 =1

(ⅲ) x1 + x2 = 2

−x1−x2 =-2

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.

n×n Systems

Definition

A system is said to be in strict triangular form if in the kth

equation the coefficients of the first k-1 variables are all

zero and the coefficient of xk is nonzero (k=1, …,n).

Example The system

is in strict triangular form.

Example Solve the system

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

2 Row Echelon Form

pivotal row

pivotal row

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.

Example Determine whether the following matrices are

in row echelon form or not.

Definition

The process of using operations Ⅰ, Ⅱ, Ⅲ to transform a linear system into one whose augmented matrix is in row echelon form is called Gaussian elimination.

Definition

A linear system is said to be overdeterminedif there are more equations than unknows.

A system of m linear equations in n unknows is said to be underdetermined if there are fewer equations than unknows (m<n).

Example

Definition

A matrix is said to be in reduced row echelon formif:

ⅰ. The matrix is in row echelon form.

ⅱ. The first nonzero entry in each row is the only nonzero entry in its column.

Homogeneous Systems

A system of linear equations is said to be homogeneous if the constants on the right-hand side are all zero.

Theorem 1.2.1An m×n homogeneous system of linear equations has a nontrivial solution if n>m.

3 Matrix Algebra

Matrix Notation

Vectors

row vector

1×n matrix

n×1 matrix

column vector

Definition

Two m×n matrices A and B are said to be equal if aij=bij for each i and j.

Scalar Multiplication

If A is a matrix andk 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×nmatrix whose (i, j) entry is kaij.

Matrix Addition

Two 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×nmatrix whose (i, j) entry is aij+bij for each ordered pair (i, j).

Example

Let

Then calculate

。

n

cij = ai1b1j + ai2b2j +…+ ainbnj = aikbkj.

k=1

Matrix Multiplication

Definition

If A=(aij) is an m×n matrix and B=(bij)is an n×rmatrix, then the product AB=C=(cij) is the m×r matrixwhose entries are defined by

Example

then calculate AB.

1. If

then calculate AB and BA.

2. If

Matrix Multiplication and Linear Systems

Case 1 One equation in Several Unknows

If we let and

then we define the product AX by

Case 2 M equations in N Unknows

If we let and

then we define the product AX by

Definition

If a1, a2, … , an are vectors in Rm and c1, c2, … , cnare 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.

- Theorem 1.3.2Each of the following statements is valid for any scalars k and l and for any matrices A, B and C for which the indicated operations are defined.
- A+B=B+A
- (A+B)+C=A+(B+C)
- (AB)C=A(BC)
- A(B+C)=AB+AC
- (A+B)+C=AC+BC
- (kl)A=k(lA)
- k(AB)=(kA)B=A(kB)
- (k+l)A=kA+lA
- k(A+B)=kA+kB

The Identity Matrix

Definition

The n×n identity is the matrix where

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.

Theorem 1.3.3If 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 defined by

bji=aij

for j=1, …, n and i=1, …, m. The transpose of A is denoted by AT.

- Algebra Rules for Transpose:
- (AT)T=A
- (kA)T=kAT
- (A+B)T=AT+BT
- (AB)T=BTAT

Definition

An n×n matrix A is said to be symmetric if AT=A.

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.

Type I. An elementary matrix of type I is a matrix obtained by

interchanging two rows ofI.

ExampleLet

and let A be a 3×3 matrix

then

Type II. An elementary matrix of type II is a matrix obtained by

multiplying a row of I by a nonzero constant.

ExampleLet

and let A be a 3×3 matrix

then

Type III. An elementary matrix of type III is a matrix obtained

from I by adding a multiple of one row to another row.

ExampleLet

and let A be a 3×3 matrix

In general, suppose that E is an n×n elementary matrix. E is obtained by either a row operation or a column operation.

If A is an n×r matrix, premultiplyingA by E has the

effect of performing that same row operation on A. If B

is an m×n matrix, postmultiplyingB by E is equivalent

to performing that same column operation on B.

Example

Let

,

Find the elementary matrices， ，such that .

Theorem 1.4.1If 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

- Theorem 1.4.2(Equivalent Conditions for Nonsingularity)
- Let A be an n×n matrix. The following are equivalent:
- A is nonsingular.
- Ax=0 has only the trivial solution 0.
- A is row equivalent to I.

Theorem 1.4.3The system of n linear equations in n unknowns Ax=b has a unique solution if and only if A is nonsingular.

A method for finding the inverse of a matrix

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 rightby A-1

EkEk-1‥‥E1I=A-1

row operations

Thus (A I)

(I A-1)

Example Compute A-1 if

Example Solve the system

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.

A is said to be triangular if it is either upper triangular or

lower triangular.

An n×n matrix A is said to be diagonal if aij=0 whenever i≠j .

5. Partitioned Matrices

- -2 4 1 3
- 1 1 1 1
- 3 2 -1 2
- 4 6 2 2 4

C11C12

=

C21 C22

C=

-1 2 1

B= 2 3 1

1 4 1

=(b1, b2, b3)

AB=A(b1, b2, b3)=(Ab1, Ab2, Ab3)

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

Then the product AB can be partitioned into rows as follows:

Block Multiplication

Let A be an m×n matrix and B an n×r matrix.

Case 1 B=(B1B2), 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))

= (AB1AB2)

Case 2 A= ,where A1 is a k×n matrix and A2

is an (m-k)×n matrix.

Thus

Case 3 A=(A1A2) 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.

Thus

Case 4 LetA and B both be partitioned as follows：

B11B12 s

B=

B21B22 n-s

t r-t

A11A12 k

A=

A21A22 m-k

s n-s

Then

In general, if the blocks have the proper dimensions, the block multiplication can be carried out in the same manner as ordinary matrix multiplication.

Example

Let

Then calculate AB.

Example

Let A be an n×n matrix of the form

where A11 is a k×k matrix (k<n). Show that

A is nonsingular if and only if A11 and A22

are nonsingular.