5.1 Rank of a Matrix | unit 5 linear algebra matrices system of linear equations

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UNIT 5

Linear Algebra-Matrices, System of Linear Equations

5.1 Rank of a Matrix

Definition:

An arrangement of m.n numbers in m rows and n columns is called a matrix of order mxn.

Generally a matrix is denoted by capital letters. Like, A, B, C, ….. etc.

2. Types of matrices:- (Review)

Row matrix
Column matrix
Square matrix
Diagonal matrix
Trace of a matrix
Determinant of a square matrix
Singular matrix
Non – singular matrix
Zero/ null matrix
Unit/ Identity matrix
Scaler matrix
Transpose of a matrix
Triangular matrices

Upper triangular and lower triangular matrices,

14. Conjugate of a matrix

15. Symmetric matrix

16. Skew – symmetric matrix

3. Operations on matrices:

Equality of two matrices
Multiplication of A by a scalar k.
Addition and subtraction of two matrices
Product of two matrices
Inverse of a matrix

4. Elementary transformations

a) Elementary row transformation

These are three elementary transformations

Interchanging any two rows (Rij)
Multiplying all elements in ist row by a non – zero constant k is denoted by KRi
Adding to the elements in ith row by the kth multiple of jth row is denoted by .

b) Elementary column transformations:

There are three elementary column transformations.

Interchanging ith and jth column. is denoted by Cij.
Multiplying ith column by a non – zero constant k is denoted by kCj.
Adding to the element of ith column by the kth multiple of jth column is denoted by Ci + kCj.

Rank of a matrix:

Let A be a given rectangular matrix or square matrix. From this matrix select any r rows from these r rows select any r columns thus getting a square matrix of order r x r. The determinant of this matrix of order r x r is called minor or order r.

e.g.

For example select 2nd and 3rd row. i.e.

Now select any two columns. Suppose, 1st and 2nd.

i.e.

5.2 System of Linear Equations

Invariance of rank through elementary transformations.

The rank of matrix remains unchanged by elementary transformations. i.e. from a matrix. A we get another matrix B by using some elementary transformation. Then

Rank of A = Rank of B

2. Equivalent matrices:

The matrix B is obtained from a matrix A by a sequence of a finite no. of elementary transformations is said to be equivalent to A. and we write.

Normal form or canonical form:

Every mxn matrix of rank r can be reduced to the form

By a finite sequence of elementary transformation. This form is called normal form or the first canonical form of the matrix A.

Q1) Reduce the following matrix to normal form of Hence find it’s rank,

S1)

We have,

Apply

Rank of A = 1

Q2) Find the rank of the matrix

S2)

We have,

Apply R12

Rank of A = 3

Q3) Find the rank of the following matrices by reducing it to the normal form.

S3)

Apply C14

H.W.

Q4) Reduce the follo9wing matrices in to the normal form and hence find their ranks.

Q5) Reduction of a matrix a to normal form PAQ.

If A is a matrix of rank r, then there exist a non – singular matrices P & Q such that PAQ is in normal form.

i.e.

To obtained the matrices P and Q we use the following procedure.

Working rule:-

If A is a mxn matrix, write A = Im A In.
Apply row transformations on A on l.h.s. and the same row transformations on the prefactor Im.
Apply column transformations on A on l.h.s and the column transformations on the postfactor In.

So that A on the l.h.s. reduces to normal form.

Ex1)

If Find Two

Matrices P and Q such that PAQ is in normal form.

S1)

Here A is a square matrix of order 3 x 3. Hence we write,

A = I3 A.I3

i.e.

Ex2)

Find a non – singular matrices p and Q such that P A Q is in normal form where

S2)

Here A is a matrix of order 3 x 4. Hence we write A as,

i.e.

5.3 Linear Dependence and Independence

Simultaneous linear equations

Non – homogeneous linear equations

Consider a system of m linear equations in n unknowns as,

…. (1)

Which can be written in matrix from as,

Ax = B … (2)

Where

The matrix [A:B] or formed by coefficient and constants is called augmented matrix.

Any set of values of the unknowns x1, x2, …..,xn which satisfied all the of the system AX = B.

When the system has at least one solution, then the system is said to be consistent, otherwise it is called inconsistent.

a) Condition of consistency:

1) If in a system of n linear equations and n unknown i.e. coefficient matrix of is of order n x n. (i.e. square matrix). & . i.e. . Then the system has no solution.

2) If coefficient matrix is of order mxn then

Then system is consistent.

b) Echelon form of a matrix:

A matrix is said to be echelon form if

1) There are some row which have all elements zero are placed at the bottom of the matrix.

2) The no. of zero elements in preceding row are less than the next row.

i.e. We have to make a given matrix. A to an upper triangular matrix by using a row transformations only.

c) Rank in echelon form

The rank of matrix is the no. of non – zero rows in echelon form of a matrix.

Ex 1) Reduce the following matrix to echelon form and find t rank.

S1)

We have

5.4 Linear and Orthogonal Transformations

Which is the required row echelon form and no. of non – zero rows are. 2.

Working rule to solve a system of non – homogeneous linear equations by reducing to echelon form.

a) Write the given system of m eqn. in n unknowns in matrix form as, AX = B.

b) Apply the row transformation on A as well as on the column matrix B. till you get an echelon from to A.

c) Then rewrite the equations as a set of linear equations.

d) We know that rank of a matrix in echelon form is equal to the no. of non – zero rows.

Case I

If rank (A) < Rank . Then equations are inconsistence i.e. they have no. solutions.

Case II

If rank A = Rank .

Then system has solution i.e. system is consistent. Further.

a) If r = n (i.e. rank of A is equal to no. of unknowns system has solution.

b) If (i.e. rank of A is less than the no. of unknowns, system has infinitely many solutions.

Ex.1)

Show that the equations

x + y = 1

2x + 3y = 1

5x – 7 = 11 are consistent and solve.

S1)

For consistency we have, Rank of A = Rank of [A l B] … (1)

Now writing the given system in matrix form as, AX = B

i.e.

R2 – 2R1, R3 – 5R1

Thus Rank of A = 2

& Rank of [A | B] = 2

system is consistent rewriting the equations as,

x + y = 1

y = -1

Solving them we get,

y = -1, x=2

Ex. 2)

Solve system by matrix method

S2)

The given system can be written in matrix form as,

Ax = B

i.e.

Here Rank of A = 23

& rank of [A | B] = 3

system has a solution further Rank of A = no. of unknowns.

it has a unique solution now rewriting the equation as,

Thus the solution is x = 2, y = 1, z = 0

Ex. 3)

Test for consistency and solve by matrix method.

S3)

The given system ca be written in matrix form as,

Ax = B

i.e.

Here Rank of A = 2

& Rank of [A | B] = 2

The system is consistent

But r < n. (i.e. 2 < 3).

Hence the system has infinitely many solutions.

Now rewrite the equations are

Put Z = t

Ex.4)

Test for consistency the following equations and if possible solve them by matrix method.

S4)

The given system can be written in matrix form as, Ax = B

i.e.

R13

Thus Rank of A = Rank of [A | B]. Hence the system is consistence now

Rewriting the equations as,

Z3 = 4

Thus solution get is

Ex. 5

For what value of ‘’ the equations have a solution and solve them completely in each case.

S5)

The given system can be written in matrix form as,

Ax = B

i.e.

… (1)

Since the system is consistent.

Case I:

If equation (1) becomes

Rewriting equation as,

Put

Case II:

For equation (1) becomes,

Rewriting equation as,

Put

Hence solution is

Ex. 6)

For what value of the equations

Will have no unique solution? Will the equations have any solution for this value of ?

S6)

The given system can be written in matrix form as, Ax = B.

R12

… (2)

The system has unique solutions if the coefficient matrix is non singular.

i.e.

Thus the above system has no unique solution for

Now if then equation (1) becomes,

Here .

Hence the system is inconsistent

Ex.7)

Investigate for what values of the equations

have

No solution
Unique solution
An infinite no. of solutions?

S7)

The given system can be written in matrix form as,

i.e.

… (1)

This system has unique solution if determinant of coefficient matrix is non – singular.

i.e.

Now if then equation (1) becomes.

… (3)

Now this system is consistent.

If and only if

Thus, if the system is consistent and it has infinitely many solutions

If then

Then system is inconsistent

Thus

No solution if

2. A Unique solution if &

3. Infinitely many solutions if

Ex.8)

Show that the system

has no solution

Unless

S8)

The given system can be written in matrix form as, Ax = 8

i.e.

Apply

R12

R3 – R2

Which is the required row echelon form,

Here

Thus the system has no solution

If and only if

i.e.

If and only if

i.e.

hence the given system has no solution unless .

Solution:

H.W.

Ex.8

Investigate for what values of the equations,

Have

No solution
A unique solution
An infinite no. of solutions.

5.5 Application to problems in Engineering

Homogeneous linear equations

If a system of linear equations every equations has r.h.s. is zero. Then this system is called homogeneous system of linear equation. Such a system can be written in matrix form as, AX = 0.

Where

A is the matrix of coefficient

X is the matrix of unknowns

O is the zero matrix (column)

Every homogeneous system has always a one solutions i.e. x1 = 0, x2 = 0, ….., xn=0. Such a solution is called as zero solute or trivial solution and all other solutions are called non–zero or non – trivial solutions (i.e. at least one of).