Chapter 1 matrix system determinant (1)

An n-by-n matrix is known as a square matrix of order n.. The entries aii form the main diagonal of a square matrix.. Upper triangular matrixA square matrix is called an upper triangular

Matrix, System of linear equations, Determinant

Phan Thi Khanh Van

E-mail: khanhvanphan@hcmut.edu.vn

May 12, 2021

Table of contents

1 Matrices

2 Elementary row (column) operations of a matrix

3 Rank of matrices

4 System of linear equations

5 The homogeneous system of linear equations

6 Operations with matrices

7 Inverse of a matrix

8 Determinant

Example

ai 1 ai 2 aij ain

Square matrix

A square matrix is a matrix with the same number ofrows and columns m = n An n-by-n matrix is known as

a square matrix of order n The set of all square matrices

of order n over the field K is denoted by Mn[K]

 - a square matrix of order 3

The entries aii form the main diagonal of a square

matrix The sum of the main diagonal a11+ a22+ + ann

is called the trace, denoted by tr (A), trace(A)

Upper triangular matrix

A square matrix is called an upper triangular matrix ifall the entries below the main diagonal are zeros

Lower triangular matrix

A square matrix is called an lower triangular matrix ifall the entries above the main diagonal are zeros

Diagonal matrix

A matrix that is both upper and lower triangular is called

a diagonal matrix: aij = 0, ∀i 6= j

Identity matrix (unit matrix)

The identity matrix of size n is the n × n square matrixwith ones on the main diagonal and zeros elsewhere:

Echelon form

A matrix is said to be in row echelon form if:

All nonzero rows are above any rows of all zeros

Each leading entry (the first non-zero number fromthe left, also called the pivot) of a row is in a column

to the right of the leading entry of the row above it.Example

Reduced row echelon form (row canonical form)

A matrix is in reduced row echelon form if:

It is in row echelon form

The leading entry in each nonzero row is 1 and is theunique nonzero entry in its column

Elementary row (column) operations of a matrix

1 (Interchange) Interchange two rows (columns): ri ↔ rj

2 (Scaling) Multiply all entries in a row (column) by anonzero constant: ri → α.ri, α 6= 0

3 (Replacement) Replace one row (column) by the sum

of itself and a multiple of another row (column)

ri → ri + α.rj, ∀α

Row equivalent

Two matrices A and B are called row equivalent:

A ∼ B (A is row equivalent to B) if B can be obtainedfrom A after a finite number of elementary row operations

Add multiples of the pivot row to each of the lowerrows, so every element in the pivot column of the

lower rows equals 0

Repeat the procedure

Reduce the matrix to the echelon form

1 The row echelon matrix that results from a series ofelementary row operations is not unique However,the reduced row echelon form is unique

2 The number of non-zero rows of any echelon matrixequals that of the reduced echelon matrix

Find the rank of

Analysis of an Electrical Network

Kirchhoff’s Laws

1 (Current Law) The

current flow into a node

equals the current flow

out of the node

2 (Voltage Law) The sum

of the RI voltage drops

in one direction around

a loop equals the sum ofthe voltage sources inthe same direction







I1 − I2 + I3 = 03I1 + 2I2 = 72I2 + 4I3 = 8

The flow of traffic through a network of streets

Find the traffic flows x1, x2, x3, x4 and x5

System of linear equations

System of m linear equations with n variables

Gaussian elimination & back substitution

We solve a linear system in 3 steps:

1 (A|b) elem row operations

2 From r (A), r (A|b), we derive the number of solutions

3 Use back substitution to find the general solution.Kronecker-Capelli theorem

The system of linear equations Ax = b, A ∈ Mm×n is

compatible if and only if r (A) = r (A|b)

r (A) < r (A|b): there is no solution

r (A) = r (A|b) = n: the solution is unique

r (A) = r (A|b) = r < n: there is an infinite number ofsolutions

The flow of traffic through a network of streets

- general solution of the system

The variables x1, x2, x4 corresponding to pivot columns

in the matrix are called basic variables

The other variables: x3, x5 are called free variables.The system has infinitely many solutions depending on 2

free parameter α, β ∈ R

The homogeneous system of linear equations

AX = 0

Remark: Such a homogeneous system AX = 0 alwayshas at least 1 solution (is consistent), namely, the trivialsolution X = (0, 0 0)T

r (A) = n: The trivial solution X = (0, 0 0) is theunique solution

r (A) < n: There is an infinite number of solutions(there are non trivial solutions)

Example of matrix multiplication

A store sells commodities c1, c2, c3 in two its branches

B1, B2 The quantities of commodities sold in B1, B2 in aweek are given in Table 1, the individual prices of

commodities are given in Table 2, the costs to the storeare given in Table 3 Find the store’s profit for a week

The quantities Q, the selling prices P, the costs C of

population now consists of 24 rabbits in the first age

class, 24 in the second, and 20 in the third How manyrabbits will be in each age class in 1 year, 2 years?

The current age distribution vector is





After 2 years the age distribution vector will be





Markov chain

Example

In a city with 1000 householders there are 3 supermarkets

A, B and C At this month, there are 200, 500 and 300

householders that go to the supermarkets A, B and C,respectively After each month, there are 10% of

customers of A change to B, 10% of those change to C;

7% of customers of B change to A, 3% of those change

to C; 8.3% of customers of C change to A, 6.7% of thosechange to B Find the numbers of customers of each

supermarket after 1 month, 2 months

The numbers of customers of A, B and C at this month





Matrix power

Let A be a square matrix The power Am for a

nonnegative integer m is the matrix product of m copies

Elementary matrices and elementary operations

1 Performing the row elementary operation on A, wewill obtain EA (E is the elementary matrix obtainedfrom the same row operation)

2 Performing the column elementary operation on A,

we will obtain AE (E is the elementary matrix

obtained from the same column operation)

The method to find A−1

A|I  −−−−−−−−−−−−−→row elementary operations I |A−1

Find the inverse of A

13 4

7 4 13

4 −114 −54

7 4

−5 4

−3 4





Application of matrix in cryptography

Chose a square matrix of order 3: K as the ”symmetrickey” and then multiply it to the left of A:

Input output Leontief model

The numbers in the table tell how much output from eachindustry a given industry requires in order to produce onedollar of its own output For example, to provide 1$ worth

of service, the service sector requires 0.04$ worth of rawmaterials, 0.03$ worth of services, and 0.01$ worth ofmanufactured goods The demand matrix D tells howmuch ( in billions of dollars) of each type of output isdemanded by consumers and others outside the economy



 Let

X = (x , y , z)T denote the production matrix It

represents the amounts (in billions of dollars of value)produced by each of the three industries

We have the equation:

Internal demand + External demand = Total production.For ex.:

+ The amount of money needed in Raw materials

industry to produce $x of Raw materials, $y of Servicesand $z of Manufacturing: 0.02x + 0.04y + 0.04z

+ The external demand of Raw materials industry: 400





a22 a23

a32 a33

+

a12.(−1)(1+2)

a21 a23

a31 a33

+ 3(−1)3+2

1 3

2 6

+ 5(−1)3+3

1 2

2 4

... 1. (? ?1) (1+ 1).

1 −2

+

3.(? ?1) (1+ 2)

1 −2

+ (? ?1) .(? ?1) (1+ 3)

2

1

= −8 + 12 + =... +

a12 .(? ?1) (1+ 2)

a 21< /small> a23

a 31< /sub> a33

+ a13 .(? ?1) (1+ 3)

... × (n − 1) matrix that resultsfrom deleting the i −th row and the j −th column of A).Determinant of an n × n matrix A

is given by the expansion:

A - n × n matrix Fixing