Module-5
Linear algebra
Question-1: 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.
Question-2: Find the rank of the following matrices by echelon form?
Let A =
Applying
A
Applying
A
Applying
A
Applying
A
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
Question-3: Find the solution of the following homogeneous system of linear equations,
Sol. 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,
Question-4: check whether the following system of linear equations is consistent of not.
2x + 6y = -11
6x + 20y – 6z = -3
6y – 18z = -1
Sol. 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.
Question-5: 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
Sol. 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.
Question-6: Solve by the Gauss-Jordan elimination method-
Sol. Here
The augmented matrix is-
Therefore, the matrix A is in normal form-
Question-7: solve the following system of linear equations by using Guass seidel method-
6x + y + z = 105
4x + 8y + 3z = 155
5x + 4y - 10z = 65
Sol. The above equations can be written as,
………………(1)
………………………(2)
………………………..(3)
Now put z = y = 0 in first eq.
We get
x = 35/2
Put x = 35/2 and z = 0 in eq. (2)
We have,
Put the values of x and y in eq. 3
Again start from eq.(1)
By putting the values of y and z
y = 85/8 and z = 13/2
We get
The process can be showed in the table format as below
At the fourth iteration , we get the values of x = 14.98 , y = 9.98 , z = 4.98
Which are approximately equal to the actual values,
As x = 15 , y = 10 and y = 5 ( which are the actual values)
Question-8: Find out the Eigen values and Eigen vectors of ?
Sol. 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).
Question-9: Find the Eigen values of Eigen vector for the matrix.
Solution:
Consider the characteristic equation as
i.e.
i.e.
are the required eigen values.
Now consider the equation
… (1)
Case I:
Equation (1) becomes,
Thus and n = 3
3 – 2 = 1 independent variables.
Now rewrite the equations as,
Put
,
I.e.
The Eigen vector for
Case II:
If equation (1) becomes,
Thus
Independent variables.
Now rewrite the equations as,
Put
Is the Eigen vector for
Now
Case III:-
If equation (1) gives,
R1 – R2
Thus
Independent variables
Now
Put
Thus
Is the Eigen vector for
Question-10: Diagonalise the matrix
Let A =
The Eigen vectors are (4,1),(1,-1) corresponding to Eigen values .
Then and also
Also we know that
Module-5
Linear algebra
Question-1: 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.
Question-2: Find the rank of the following matrices by echelon form?
Let A =
Applying
A
Applying
A
Applying
A
Applying
A
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
Question-3: Find the solution of the following homogeneous system of linear equations,
Sol. 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,
Question-4: check whether the following system of linear equations is consistent of not.
2x + 6y = -11
6x + 20y – 6z = -3
6y – 18z = -1
Sol. 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.
Question-5: 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
Sol. 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.
Question-6: Solve by the Gauss-Jordan elimination method-
Sol. Here
The augmented matrix is-
Therefore, the matrix A is in normal form-
Question-7: solve the following system of linear equations by using Guass seidel method-
6x + y + z = 105
4x + 8y + 3z = 155
5x + 4y - 10z = 65
Sol. The above equations can be written as,
………………(1)
………………………(2)
………………………..(3)
Now put z = y = 0 in first eq.
We get
x = 35/2
Put x = 35/2 and z = 0 in eq. (2)
We have,
Put the values of x and y in eq. 3
Again start from eq.(1)
By putting the values of y and z
y = 85/8 and z = 13/2
We get
The process can be showed in the table format as below
At the fourth iteration , we get the values of x = 14.98 , y = 9.98 , z = 4.98
Which are approximately equal to the actual values,
As x = 15 , y = 10 and y = 5 ( which are the actual values)
Question-8: Find out the Eigen values and Eigen vectors of ?
Sol. 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).
Question-9: Find the Eigen values of Eigen vector for the matrix.
Solution:
Consider the characteristic equation as
i.e.
i.e.
are the required eigen values.
Now consider the equation
… (1)
Case I:
Equation (1) becomes,
Thus and n = 3
3 – 2 = 1 independent variables.
Now rewrite the equations as,
Put
,
I.e.
The Eigen vector for
Case II:
If equation (1) becomes,
Thus
Independent variables.
Now rewrite the equations as,
Put
Is the Eigen vector for
Now
Case III:-
If equation (1) gives,
R1 – R2
Thus
Independent variables
Now
Put
Thus
Is the Eigen vector for
Question-10: Diagonalise the matrix
Let A =
The Eigen vectors are (4,1),(1,-1) corresponding to Eigen values .
Then and also
Also we know that