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Chapter 1. Matrices and Systems of Equations. Systems of Linear Equations. Where the a ij ’ s and b i ’ s are all real numbers, x i ’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.

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Chapter 1

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Chapter 1

Chapter 1

Matrices and Systems of Equations


Chapter 1

  • 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.


Chapter 1

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


Chapter 1

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.


Chapter 1

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.


Chapter 1

Example Solve the system


Chapter 1

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


Chapter 1

2 Row Echelon Form

pivotal row

pivotal row


Chapter 1

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.


Chapter 1

Example Determine whether the following matrices are

in row echelon form or not.


Chapter 1

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).


Chapter 1

Example


Chapter 1

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.


Chapter 1

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.


Chapter 1

3 Matrix Algebra

Matrix Notation


Chapter 1

Vectors

row vector

1×n matrix

n×1 matrix

column vector


Chapter 1

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.


Chapter 1

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).


Chapter 1

Example

Let

Then calculate


Chapter 1

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


Chapter 1

Example

then calculate AB.

1. If

then calculate AB and BA.

2. If


Chapter 1

Matrix Multiplication and Linear Systems

Case 1 One equation in Several Unknows

If we let and

then we define the product AX by


Chapter 1

Case 2 M equations in N Unknows

If we let and

then we define the product AX by


Chapter 1

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.


Chapter 1

  • 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


Chapter 1

The Identity Matrix

Definition

The n×n identity is the matrix where


Chapter 1

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.


Chapter 1

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.


Chapter 1

  • 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.


Chapter 1

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.


Chapter 1

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


Chapter 1

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


Chapter 1

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


Chapter 1

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.


Chapter 1

Example

Let

,

Find the elementary matrices, ,such that .


Chapter 1

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


Chapter 1

  • 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.


Chapter 1

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)


Chapter 1

Example Compute A-1 if


Chapter 1

Example Solve the system


Chapter 1

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 .


Chapter 1

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)


Chapter 1

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:


Chapter 1

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)


Chapter 1

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


Chapter 1

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


Chapter 1

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


Chapter 1

Example

Let

Then calculate AB.


Chapter 1

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.


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