Unit – 6

Content

6.1 Elementary row operations in matrix

6.2 Row echelon and reduced row echelon froms

6.3 Rank by echelon form

6.4 Inverse by gauss – Jordan method

6.5 Solution of system of linear equation by gauss elimination and gauss Jordan method

6.6 Eigen values and Eigen vector

6.7 Cayley – Hamilton theorem

6.8 Diagonalization  of a matrix

6.1 Elementary row operations in matrix

Matrices: It is a rectangular arrangement of numbers formed by keeping them in m numbers of rows and n numbers of columns.

Algebra on Matrices:

1. Addition and subtraction of matrices:

Addition and subtraction of matrices is possible if and only if they are of same order.

We add or subtract the corresponding elements of the matrices.

Example:

2.     Scalar multiplication of matrix:

In this we multiply the scalar or constant with each element of the matrix.

Example:

3.     Multiplication of matrices: Two matrices can be multiplied only if they are conformal i.e. the number of column of first matrix is equal to the number rows of the second matrix.

Example:

Then

4.     Power of Matrices: If A is A square matrix then

  and so on.

If where A is square matrix then it is said to be idempotent.

5. Transpose of  a  matrix: The  matrix  obtained  from  any  given  matrix A ,  by interchanging rows  and   columns is  called  the  transpose of  A and is  denoted  by

The transpose of matrix  Also 

Note:

Special  Matrix:

Orthogonal   Matrices:

A real square matrix A is said to be orthogonal if 

Since .

Taking determinant we get  

As   we know that

So,.

This implies that determinant of an orthogonal   matrix is either 1 or -1.

Note: i) If A   is orthogonal, A’   and inverse   of A is also orthogonal.

Ii) If A and B   are orthogonal then AB is also orthogonal.

Example1:   Prove that the following matrix is orthogonal:

Let A  =   then  A’= 

Now,   AA’=

Thus AA’=   = I

Hence the matrix is orthogonal.

Singular Matrix: A square matrix is said to be singular   if its determinant is zero otherwise non singular matrix.

i.e. |A|=0 then A is a singular matrix if not non singular.

1.1 Elementary transformation of matrices

It contains three operations one on the rows or columns of the matrix:

1. Interchanging- It contain interchange of any two rows or columns .i.e.

b.     Scalar multiplication-It contains scalar multiplication of any scalar quantity with any row or column.

c.      Linear combination: we form a linear combination by multiplying a scalar with any row or column and then add the corresponding elements with another row or column.

t

NOTE: Elementary transformation does not change/affect the order or rank of the matrix.

Elementary matrix: It is obtained from an identity matrix by applying any elementary transformation to it

6.2 Row echelon and reduced row echelon forms

Row echelon from-

Any matrix is said to be in row echelon form if it follows the term mentioned below:

1. The first non-zero element in each row , is called leading coefficient , is 1. (  some prefers it is not necessary that it should be 1. It can be any number)

2. Each leading entry is in a column to the right of the leading entry in the previous row.

3. Rows with all zero elements are below rows having a non-zero element.

Example of a row echelon form-

      and      

Row reduced form-

A matrix is said to be in reduced row echelon form if it satisfies the following terms-

1. The first non-zero entry in each row is 1.

2. The columns containing this 1 has all the other entries 0.

A matrix in row reduced form is known as row reduced matrix.

For example:

The above matrices are row reduced matrix.

Important definition-

Leading term- the first non-zero entry of any row is called leading term.

Leading column- the column containing the leading term is known as leading column.

Row reduced echelon from-

A matrix is said to be in row reduced echelon form if it has following properties,

1. If the given matrix is already in row reduced form.

2. The row consisting of all zeroes comes below all non zero rows

3. The leading term appear from left to right in successive rows

Some of the examples of row reduced echelon form of the matrices are given below,

6.3 Rank by echelon form

Rank of a matrix-

The rank of a matrix (r) can be defined as –

1. It has atleast one non-zero minor of order r.

2. Every minor of A of order higher than r is zero.

Example: Find the rank of a matrix M by echelon form.

M = 

Sol. First we will convert the matrix M into echelon form,

M =

Apply,    , we get

M =

Apply  , we get

M =

Apply 

M =

We can see that, in this echelon form of matrix, the number of non – zero rows is 3.

So that the rank of matrix X will be 3.

Example: Find the rank of a matrix A by echelon form.

A =

Sol. Convert the matrix A into echelon form,

A =

Apply 

A =

Apply   , we get

A =

Apply , we get

A =

Apply ,

A =

Apply  ,

A =

Therefore the rank of the matrix will be 2.

Example:  Find the rank of a matrix A by echelon form.

A =

Sol. Transform the matrix A into echelon form, then find the rank,

We have,

A =

Apply, 

A =

Apply ,

A =

Apply

A =

Apply

A =

Hence the rank of the matrix will be 2.

6.4 Inverse by gauss – Jordan method

This can be done with the help of Gauss-Jordan method. This method used only elementary row transformation, with the help of it we convert the square matrix A to the identity matrix I and the corresponding matrix obtained give the inverse of A.

Gauss –Jordan Method of finding the inverse:

We know that X will be the inverse of a matrix A if

Where I is an identity matrix of order same as A.

We write, using elementary operation we covert the matrix A in to a upper triangular matrix. Then compare this matrix with each corresponding column and using back substitution.

The determinant of this coefficient matrix is the product of the diagonal elements of the upper triangular matrix.

If this |A|=0 then inverse does not exist.

Example1: Find the inverse of the matrix   by row transformation?

Let  A=

By Gauss-Jordan method

We have

=

Apply

Apply we get

Apply we get

Apply

Apply

Hence

Example2: Find the inverse of

Let A=

By Gauss-Jordan method

We have

=

Apply

Apply

Apply

Apply

Hence

Example3: Find the inverse of the matrix   by row transformation?

Let A=

By Gauss-Jordan method

We have

=

Apply we get

Apply

Apply

Apply

Apply

Apply

Hence

Example4: Find the inverse of 

Let A=

By Gauss Jordan method

Apply

Apply

Apply 

Apply

Hence the inverse of matrix A is

Example5:Find the inverse of

Let A=

By Gauss-Jordan Method [A:I]

=

Apply

Apply 

Apply

Apply

Apply

Hence the inverse of matrix A is

6.5 Solution of system of linear equation by gauss elimination and gauss Jordan method

The matrix that we make from the coefficient x,y and z of of the linear equations is reduced into echelon form by elementary row operation.

At the end, we calculate the value of z, and value of y and x are calculated by backward substitution.

Example: find the solution of the following linear equations.

+ - = 1

- + = 2

- 2 + = 5

Sol. These equations can be converted into the form of matrix as below-

Apply the operation, and     , we get

We get the following set of equations from the above matrix,

+ - = 1  …………………..(1)

+ 5 = -1  …………………(2)

= k

Put x = k  in eq. (1)

We get,

+ 5k = -1

Put these values in eq. (1), we get

     and  

These equations have infinitely many solutions.

Guass- Jordan method –

This method is the modification of gauss elimination method

Example: solve the following system of equations by gauss-jordan method

+ - 2 = 0

+ + = 0

- 7 + = 0

Sol. The given system of equations can be written in the form of matrices as follows,

By applying operation  -     we get,

Now apply,   

We get,

Apply,   

The set of the equations we get from above matrices,

+ + = 0 …………………..(1)

21 + = 0  …………………………..(2)

Suppose

From equation-2, we get

21

Now from equation-1:

- ()  + 4b + 9a = 0

We get,

Example: solve the following system of equations by gauss-jordan method

2x + y + 2z = 10

X + 2y + z = 8

3x + y – z = 2

Sol.  We will write the equations in matrix form as follows,

Now apply operation, 

Apply  , we get,

Apply,    , we get,

Apply,  / -3 , we get

Apply

Apply / -4, we get

Finally apply                   

Therefore the solution of the set of linear equations will be,

x = 1 , y = 2 , z = 3

Example:  solve the following system of equations by gauss-jordan method.

6a + 8b +6c + 3d = -3

6a – 8b + 6c – 3d = 3

8b           - 6d = 6

Sol. Change the system of linear equations into matrix form,

The augmented matrix format will be,

Apply 

Apply

Apply 

Apply    and

Apply 

Apply   

This is the reduced row echelon form,

The system of linear equations becomes,

a + c = 0

b = 0

d = -1

Suppose c = t be a free variable,

Then the solution will be, 

a = -t

b = 0

c = t

d = -1

For any number ‘t’.

6.6 Eigen values and Eigen vector

Let A is a square matrix of order n. The equation formed by

Where I is an identity matrix of order n and is unknown. It is called characteristic equation of the matrix A.

The values of the are called the root of the characteristic equation, they are also known as characteristics roots or latent root or Eigen values of the matrix A.

Corresponding to each Eigen value there exist vectors X,

Called the characteristics vectors or latent vectors or Eigen vectors of the matrix A.

Note:   Corresponding to distinct  Eigen value we get distinct Eigen vectors but in case of repeated Eigen values  we can have or not  linearly independent Eigen vectors.

If is Eigen vectors corresponding to Eigen value then is also Eigen vectors for scalar c.

Properties of Eigen Values:

1. The  sum  of the  principal diagonal element of the matrix is equal to the sum of the all Eigen values of the matrix.

Let A be a matrix of order 3 then

2.     The determinant of the matrix A is equal to the product of the all Eigen values of the matrix then .

3.     If is the Eigen value of the matrix A then 1/ is the Eigen value of the .

4.     If is the Eigen value of an orthogonal matrix, then 1/ is also its Eigen value.

5.     If are the Eigen values of the matrix A then has the Eigen values .

Example1: Find the sum and the product of the Eigen values of ?

The sum of Eigen values =  the sum of the diagonal elements

   =1+(-1)=0

The product of the Eigen values is the determinant of the matrix

On solving above equations we get

Example2:  Find out the Eigen values and Eigen vectors of ?

The Characteristics equation is given by

Or

Hence the Eigen values are 0,0 and 3.

The Eigen vector corresponding to Eigen value is

Where X is the column matrix of order 3 i.e.

This implies that

Here number of unknowns are 3 and number of equation is 1.

Hence we have (3-1)=2 linearly independent solutions.

Let

Thus the Eigen vectors corresponding to the Eigen value are (-1,1,0) and (-2,1,1).

The Eigen vector corresponding to Eigen value is

Where X is the column matrix of order 3 i.e.

This implies that

Taking last two equations we get

Or

Thus the Eigen vectors corresponding to the Eigen value are (3,3,3).

Hence the three Eigen vectors obtained are  (-1,1,0), (-2,1,1) and (3,3,3).

Example3: Find out the Eigen values and Eigen vectors of

Let A =

The characteristics equation of A is .

Or 

Or 

Or

Or

The Eigen vector corresponding to Eigen value is

Where X is the column matrix of order 3 i.e.

Or

On solving we get 

Thus the Eigen vectors corresponding to the Eigen value is (1,1,1).

The Eigen vector corresponding to Eigen value is

Where X is the column matrix of order 3 i.e.

Or    

On solving or .

Thus the Eigen vectors corresponding to the Eigen value is (0,0,2).

The Eigen vector corresponding to Eigen value is

Where X is the column matrix of order 3 i.e.

Or    

On solving we get    or .

Thus the Eigen vectors corresponding to the Eigen value is (2,2,2).

Hence three Eigen vectors are (1,1,1), (0,0,2) and (2,2,2).

6.7 Cayley – Hamilton theorem

It states that every square matrix A when substituted in its characteristics equation, will satisfies it.

Let A is a square matrix of order n. The characteristic equation of the matrix A.

Then according to Cayley-Hamilton theorem

We can also find the inverse of A ,

Multiplying on both side of above equation we get

Or

Example1:  Verify the Cayley-Hamilton theorem and find the inverse.

?

Let A =

The characteristics equation of A is

Or

Or 

Or 

By Cayley-Hamilton theorem 

L.H.S:

=   =0=R.H.S

Multiply both side by on 

Or

Or [

Or

Example2: Verify the Cayley-Hamilton theorem and find the inverse.

The characteristics equation of A is

Or

Or

Or

Or

Or

By Cayley-Hamilton theorem

L.H.S.

=

=

=

Multiply both side with in

Or

Or

=

Example3: Using Cayley-Hamilton theorem, find , if A = ?

Let A =

The characteristics equation of A is

Or

Or

By Cayley-Hamilton theorem   

L.H.S.

=

By Cayley-Hamilton theorem we have   

Multiply both side by

.

Or 

=

=

6.8 Diagonalization  of a matrix

Two square matrix and A of same order n are said to be similar if and only if

for some non singular matrix P.

Such transformation of the matrix A into with the help of non singular matrix P is known as similarity transformation.

Similar matrices have the same Eigen values.

If X is an Eigen vector of matrix A then is Eigen vector of the matrix  

Reduction to Diagonal Form:

Let A be  a square matrix of order n has n linearly independent Eigen vectors  which form the  matrix P such that

Where P is called the modal matrix and D is known as spectral matrix.

Procedure: let A be a square   matrix of order 3.

Let three Eigen vectors of A are  corresponding  to  Eigen values

Let     

  {by characteristics equation of A}

Or

Or 

Note: The method of diagonalization is helpful in calculating power of a matrix.

  .Then for an integer n we have  

We are using the example of 1.6*

Example1:  Diagonalise the matrix

Let A=

The three Eigen vectors obtained are  (-1,1,0), (-1,0,1) and (3,3,3) corresponding to Eigen values .

Then    and

Also we know that    

Example2: Diagonalise the matrix

Let A =

The Eigen vectors are (4,1),(1,-1) corresponding to Eigen values .

Then    and also 

Also we know that