Unit – 4 | unit 4 matrices

M-I

Unit – 4

Matrices

Q1) What do you understand by elementary transformation?

A1)

Elementary transformation

A matrix can be transformed to another matrix by performing certain operations, we call these operations as elementary row operations and elementary column operations.

Definition-

“Two matrices A and B of same order are said to be equivalent to one another if matrix which need not be equal to the given matrix.

Elementary row and column operations-

The interchanging of any two rows or columns of the matrix
Replacing a row or column of the matrix by a non-zero scalar multiple of the row (column) by a non-zero scalar.
Replacing a row or column of the matrix by a sum of the row (column) with a non-zero scalar multiple of another row or column of the matrix.

One can be obtained from the other by the applications of elementary transformations.”

Q2) Reduce the following matrix into row-echelon form-

A2)

We have-

Apply-

Q3) Find the rank of a matrix A by echelon form.

A =

A3)

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.

Q4) Find the rank of the following matrices by echelon form?

A4)

Let A =

Applying

It is clear that minor of order 3 vanishes but minor of order 2 exists as

Hence rank of a given matrix A is 2 denoted by

Q5) Example) Reduce the following matrix to normal form of Hence find it’s rank,

A5)

We have,

Apply

Rank of A = 1

Q6) Find the rank of the following matrices by reducing it to the normal form.

A6)

Apply C14

Q7) Find the solution of the following homogeneous system of linear equations,

A7)

The given system of linear equations can be written in the form of matrix as follows,

Apply the elementary row transformation,

, we get,

, we get

Here r(A) = 4, so that it has trivial solution,

Q8) Check whether the following system of linear equations is consistent of not.

2x + 6y = -11

6x + 20y – 6z = -3

6y – 18z = -1

A8)

Write the above system of linear equations in augmented matrix form,

Apply , we get

Apply

Here the rank of C is 3 and the rank of A is 2

Therefore both ranks are not equal. So that the given system of linear equations is not consistent.

Q9) Check the consistency and find the values of x , y and z of the following system of linear equations.

2x + 3y + 4z = 11

X + 5y + 7z = 15

3x + 11y + 13z = 25

A9)

Re-write the system of equations in augmented matrix form.

C = [A,B]

That will be,

Apply

Now apply ,

We get,

Here rank of A = 3

And rank of C = 3, so that the system of equations is consistent,

So that we can can solve the equations as below,

That gives,

x + 5y + 7z = 15 ……………..(1)

y + 10z/7 = 19/7 ………………(2)

4z/7 = 16/7 ………………….(3)

From eq. (3)

z = 4,

From 2,

From eq.(1), we get

x + 5(-3) + 7(4) = 15

That gives,

x = 2

Therefore the values of x , y , z are 2 , -3 , 4 respectively.

Q10) Are the vectors , , linearly dependent. If so, express x1 as a linear combination of the others.

A10)

Consider a vector equation,

i.e.

Which can be written in matrix form as,

Here & no. Of unknown 3. Hence the system has infinite solutions. Now rewrite the questions as,

Put

and

Thus

i.e.

Since F11 k2, k3 not all zero. Hence are linearly dependent.

Q11) At what value of P the following vectors are linearly independent.

A11)

Consider the vector equation.

i.e.

This is a homogeneous system of three equations in 3 unknowns and has a unique trivial solution.

If and only if Determinant of coefficient matrix is non zero.

consider .

i.e.

Thus for the system has only trivial solution and Hence the vectors are linearly independent.

Q12) Determine the eigen values of eigen vector of the matrix.

A12)

Consider the characteristic equation as,

i.e.

Which is the required characteristic equation.

are the required eigen values.

Now consider the equation

… (1)

Case I:

If Equation (1) becomes

R1 + R2

Thus

independent variable.

Now rewrite equation as,

Put x3 = t

Thus .

Is the eigen vector corresponding to .

Case II:

If equation (1) becomes,

Here

independent variables

Now rewrite the equations as,

Put

Is the eigen vector corresponding to .

Case III:

If equation (1) becomes,

Here rank of

independent variable.

Now rewrite the equations as,

Put

Thus .

Is the eigen vector for .

Q13) What is the diagonalization of square 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.

Q14) Diagonalize the matrix

A14)

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

Q15) State Cayley-Hamilton theorem.

A15)

Every square matrix satisfies its characteristic equation, that means for every square matrix of order n,

|A - | =

Then the matrix equation-

Is satisfied by X = A

That means

Q16) Find the characteristic equation of the matrix A = and verify cayley-Hamlton theorem.

A16)

Characteristic equation of the matrix, we can be find as follows-

Which is,

( 2 - , which gives

According to cayley-Hamilton theorem,

…………(1)

Now we will verify equation (1),

Put the required values in equation (1) , we get

Hence the cayley-Hamilton theorem is verified.

Q17) Using Cayley-Hamilton theorem, find , if A = ?

A17)

Let A =

The characteristics equation of A is

By Cayley-Hamilton theorem

L.H.S.

By Cayley-Hamilton theorem we have

Multiply both side by

Q18) Find the inverse of matrix A by using Cayley-Hamilton theorem.

A =

A18)

The characteristic equation will be,

|A - | = 0

Which gives,

(4-

According to Cayley-Hamilton theorem,

Multiplying by

That means

On solving,

So that,

Q19) Find the inverse of matrix A by using Cayley-Hamilton theorem.

A =

A19)

The characteristic equation will be,

|A - | = 0

= (2-

= (2 -

That is,

We know that by Cayley-Hamilton theorem,

…………………….(1)t,

Multiply equation(1) by , we get

Now we will find

Hence the inverse of matrix A is,

Q20) Define inner product spaces.

A20)

Let V be a vector space over F. An inner product on V

Is a function that assigns, to every ordered pair of vectors x and y in V, a scalar in F, denoted , such that for all x, y, and z in V and all c in F, the following hold:

(a)

(b)

(c)

(d)

Note that (c) reduces to = if F = R. Conditions (a) and (b) simply require that the inner product be linear in the first component.

It is easily shown that if and , then

Q21) What do you understand by Gram-Schmidt orthogonalization

A21)

Suppose {} is a basis of an inner product space V. One can use this basis to construct an orthogonal basis {} of V as follows. Set

………………

……………….

In other words, for k = 2, 3, . . . , n, we define

Where

Is the component of .

Each is orthogonal to the preceeding w’s. Thus, form an orthogonal basis for V as claimed. Normalizing each wi will then yield an orthonormal basis for V.

Unit – 4

Matrices

Q1) What do you understand by elementary transformation?

A1)

Elementary transformation

A matrix can be transformed to another matrix by performing certain operations, we call these operations as elementary row operations and elementary column operations.

Definition-

“Two matrices A and B of same order are said to be equivalent to one another if matrix which need not be equal to the given matrix.

Elementary row and column operations-

The interchanging of any two rows or columns of the matrix
Replacing a row or column of the matrix by a non-zero scalar multiple of the row (column) by a non-zero scalar.
Replacing a row or column of the matrix by a sum of the row (column) with a non-zero scalar multiple of another row or column of the matrix.

One can be obtained from the other by the applications of elementary transformations.”

Q2) Reduce the following matrix into row-echelon form-

A2)

We have-

Apply-

Q3) Find the rank of a matrix A by echelon form.

A =

A3)

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.

Q4) Find the rank of the following matrices by echelon form?

A4)

Let A =

Applying

It is clear that minor of order 3 vanishes but minor of order 2 exists as

Hence rank of a given matrix A is 2 denoted by

Q5) Example) Reduce the following matrix to normal form of Hence find it’s rank,

A5)

We have,

Apply

Rank of A = 1

Q6) Find the rank of the following matrices by reducing it to the normal form.

A6)

Apply C14

Q7) Find the solution of the following homogeneous system of linear equations,

A7)

The given system of linear equations can be written in the form of matrix as follows,

Apply the elementary row transformation,

, we get,

, we get

Here r(A) = 4, so that it has trivial solution,

Q8) Check whether the following system of linear equations is consistent of not.

2x + 6y = -11

6x + 20y – 6z = -3

6y – 18z = -1

A8)

Write the above system of linear equations in augmented matrix form,

Apply , we get

Apply

Here the rank of C is 3 and the rank of A is 2

Therefore both ranks are not equal. So that the given system of linear equations is not consistent.

Q9) Check the consistency and find the values of x , y and z of the following system of linear equations.

2x + 3y + 4z = 11

X + 5y + 7z = 15

3x + 11y + 13z = 25

A9)

Re-write the system of equations in augmented matrix form.

C = [A,B]

That will be,

Apply

Now apply ,

We get,

Here rank of A = 3

And rank of C = 3, so that the system of equations is consistent,

So that we can can solve the equations as below,

That gives,

x + 5y + 7z = 15 ……………..(1)

y + 10z/7 = 19/7 ………………(2)

4z/7 = 16/7 ………………….(3)

From eq. (3)

z = 4,

From 2,

From eq.(1), we get

x + 5(-3) + 7(4) = 15

That gives,

x = 2

Therefore the values of x , y , z are 2 , -3 , 4 respectively.

Q10) Are the vectors , , linearly dependent. If so, express x1 as a linear combination of the others.

A10)

Consider a vector equation,

i.e.

Which can be written in matrix form as,

Here & no. Of unknown 3. Hence the system has infinite solutions. Now rewrite the questions as,

Put

and

Thus

i.e.

Since F11 k2, k3 not all zero. Hence are linearly dependent.

Q11) At what value of P the following vectors are linearly independent.

A11)

Consider the vector equation.

i.e.

This is a homogeneous system of three equations in 3 unknowns and has a unique trivial solution.

If and only if Determinant of coefficient matrix is non zero.

consider .

i.e.

Thus for the system has only trivial solution and Hence the vectors are linearly independent.

Q12) Determine the eigen values of eigen vector of the matrix.

A12)

Consider the characteristic equation as,

i.e.

Which is the required characteristic equation.

are the required eigen values.

Now consider the equation

… (1)

Case I:

If Equation (1) becomes

R1 + R2

Thus

independent variable.

Now rewrite equation as,

Put x3 = t

Thus .

Is the eigen vector corresponding to .

Case II:

If equation (1) becomes,

Here

independent variables

Now rewrite the equations as,

Put

Is the eigen vector corresponding to .

Case III:

If equation (1) becomes,

Here rank of

independent variable.

Now rewrite the equations as,

Put

Thus .

Is the eigen vector for .

Q13) What is the diagonalization of square 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.

Q14) Diagonalize the matrix

A14)

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

Q15) State Cayley-Hamilton theorem.

A15)

Every square matrix satisfies its characteristic equation, that means for every square matrix of order n,

|A - | =

Then the matrix equation-

Is satisfied by X = A

That means

Q16) Find the characteristic equation of the matrix A = and verify cayley-Hamlton theorem.

A16)

Characteristic equation of the matrix, we can be find as follows-

Which is,

( 2 - , which gives

According to cayley-Hamilton theorem,

…………(1)

Now we will verify equation (1),

Put the required values in equation (1) , we get

Hence the cayley-Hamilton theorem is verified.

Q17) Using Cayley-Hamilton theorem, find , if A = ?

A17)

Let A =

The characteristics equation of A is

By Cayley-Hamilton theorem

L.H.S.

By Cayley-Hamilton theorem we have

Multiply both side by

Q18) Find the inverse of matrix A by using Cayley-Hamilton theorem.

A =

A18)

The characteristic equation will be,

|A - | = 0

Which gives,

(4-

According to Cayley-Hamilton theorem,

Multiplying by

That means

On solving,

So that,

Q19) Find the inverse of matrix A by using Cayley-Hamilton theorem.

A =

A19)

The characteristic equation will be,

|A - | = 0

= (2-

= (2 -

That is,

We know that by Cayley-Hamilton theorem,

…………………….(1)t,

Multiply equation(1) by , we get

Now we will find

Hence the inverse of matrix A is,

Q20) Define inner product spaces.

A20)

Let V be a vector space over F. An inner product on V

Is a function that assigns, to every ordered pair of vectors x and y in V, a scalar in F, denoted , such that for all x, y, and z in V and all c in F, the following hold:

(a)

(b)

(c)

(d)

Note that (c) reduces to = if F = R. Conditions (a) and (b) simply require that the inner product be linear in the first component.

It is easily shown that if and , then

Q21) What do you understand by Gram-Schmidt orthogonalization

A21)

Suppose {} is a basis of an inner product space V. One can use this basis to construct an orthogonal basis {} of V as follows. Set

………………

……………….

In other words, for k = 2, 3, . . . , n, we define

Where

Is the component of .

Each is orthogonal to the preceeding w’s. Thus, form an orthogonal basis for V as claimed. Normalizing each wi will then yield an orthonormal basis for V.

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