Unit – 5
Linear Equations and Matrix Theory
Q1) What do you understand by a matrix?
A1)
A matrix is a rectangular arrangement of the numbers.
These numbers inside the matrix are known as elements of the matrix.
A matrix ‘A’ is expressed as-
The vertical elements are called columns and the horizontal elements are rows of the matrix.
The order of matrix A is m by n or (m× n)
Q2) What do you understand by the transpose of a matrix?
A2)
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 
Q3) What is the trace of a matrix?
A3)
Suppose A be a square matrix, then the sum of its diagonal elements is known as trace of the matrix.
Example- If we have a matrix A-
Then the trace of A = 0 + 2 + 4 = 6
Q4) Find the rank of a matrix M by echelon form.
M = 
A4)
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.
Q5) Find the rank of a matrix A by echelon form.
A = 
A5)
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.
Q6) What is characteristic equation?
A6)
Suppose we have A = 
be an n × n matrix , then
Characteristic equation- The equation |
= 0 is called the characteristic equation of A , where |
is called characteristic matrix of A. Here I is the identity matrix.
The determinant |
is called the characteristic polynomial of A.
Characteristic roots-the roots of the characteristic equation are known as characteristic roots or Eigen values or characteristic values.
Q7) Find the characteristic equation of the matrix A:
A = 
A7)
The characteristic equation will be-
|
= 0
= 0
On solving the determinant, we get
(4-
Or 
On solving we get,
Which is the characteristic equation of matrix A.
Q8) Find the characteristic equation and characteristic roots of the matrix A:
A = 
A8)
We know that the characteristic equation of the matrix A will be-
|
= 0
So that matrix A becomes,
= 0
Which gives , on solving
(1-
= 0
Or 
Or (
Which is the characteristic equation of matrix A.
The characteristic roots will be,
(
(
(
(
Values of 
are-
These are the characteristic roots of matrix A.
Q9) Find the characteristic equation and characteristic roots of the matrix A:
A = 
A9)
We know that, the characteristic equation is-
|
= 0
= 0
Which gives,
(1-
Characteristic roots are- 
Q10) What do you understand by Eigen values and Eigen vectors?
A10)
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.
Q11) Write down the properties of Eigen values.
A11)
Properties of Eigen Values:
- 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.
If 
are the Eigen values of the matrix A then 
has the Eigen values 
Q12) Find the sum and the product of the Eigen values of 
?
A12)
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 
Q13) Find out the Eigen values and Eigen vectors of 
?
A13)
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).
Q14) Find out the Eigen values and Eigen vectors of 
A14)
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).
Q15) Check whether the following system of linear equations is consistent of not.
2x + 6y = -11
6x + 20y – 6z = -3
6y – 18z = -1
A15)
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
Q16) The mapping 
defined by-
Is a linear transformation 
.
What is the kernel of this linear transformation.
A16)
Let 
be any two elements of 
Let a, b be any two elements of F.
We have
(
= 
So that f is a linear transformation.
To show that f is onto 
. Let 
be any elements 
.
Then 
and we have 
So that f is onto 
Therefore f is homomorphism of 
onto 
.
If W is the kernel of this homomorphism then 
We have
∀
Also if 
then 
Implies 
Therefore 
Hence W is the kernel of f.
Q17) 
be the linear operator defined by F(x, y) = (2x + 3y, 4x – 5y).
Find the matrix representation of F relative to the basis S = { 
} = {(1, 2), (2, 5)}
A17)
(1) First find F(
, and then write it as a linear combination of the basis vectors 
and 
. (For notational convenience, we use column vectors.) We have
And
X + 2y = 8
2x + 5y = -6
Solve the system to obtain x = 52, y =-22. Hence,
Now
X + 2y = 19
2x + 5y = -17
Solve the system to obtain x = 129, y =-55. Hence,
Now write the coordinates of 
and 
as columns to obtain the matrix
Unit – 5
Linear Equations and Matrix Theory
Q1) What do you understand by a matrix?
A1)
A matrix is a rectangular arrangement of the numbers.
These numbers inside the matrix are known as elements of the matrix.
A matrix ‘A’ is expressed as-
The vertical elements are called columns and the horizontal elements are rows of the matrix.
The order of matrix A is m by n or (m× n)
Q2) What do you understand by the transpose of a matrix?
A2)
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 
Q3) What is the trace of a matrix?
A3)
Suppose A be a square matrix, then the sum of its diagonal elements is known as trace of the matrix.
Example- If we have a matrix A-
Then the trace of A = 0 + 2 + 4 = 6
Q4) Find the rank of a matrix M by echelon form.
M = 
A4)
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.
Q5) Find the rank of a matrix A by echelon form.
A = 
A5)
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.
Q6) What is characteristic equation?
A6)
Suppose we have A = 
be an n × n matrix , then
Characteristic equation- The equation |
= 0 is called the characteristic equation of A , where |
is called characteristic matrix of A. Here I is the identity matrix.
The determinant |
is called the characteristic polynomial of A.
Characteristic roots-the roots of the characteristic equation are known as characteristic roots or Eigen values or characteristic values.
Q7) Find the characteristic equation of the matrix A:
A = 
A7)
The characteristic equation will be-
|
= 0
= 0
On solving the determinant, we get
(4-
Or 
On solving we get,
Which is the characteristic equation of matrix A.
Q8) Find the characteristic equation and characteristic roots of the matrix A:
A = 
A8)
We know that the characteristic equation of the matrix A will be-
|
= 0
So that matrix A becomes,
= 0
Which gives , on solving
(1-
= 0
Or 
Or (
Which is the characteristic equation of matrix A.
The characteristic roots will be,
(
(
(
(
Values of 
are-
These are the characteristic roots of matrix A.
Q9) Find the characteristic equation and characteristic roots of the matrix A:
A = 
A9)
We know that, the characteristic equation is-
|
= 0
= 0
Which gives,
(1-
Characteristic roots are- 
Q10) What do you understand by Eigen values and Eigen vectors?
A10)
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.
Q11) Write down the properties of Eigen values.
A11)
Properties of Eigen Values:
- 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.
If 
are the Eigen values of the matrix A then 
has the Eigen values 
Q12) Find the sum and the product of the Eigen values of 
?
A12)
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 
Q13) Find out the Eigen values and Eigen vectors of 
?
A13)
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).
Q14) Find out the Eigen values and Eigen vectors of 
A14)
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).
Q15) Check whether the following system of linear equations is consistent of not.
2x + 6y = -11
6x + 20y – 6z = -3
6y – 18z = -1
A15)
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
Q16) The mapping 
defined by-
Is a linear transformation 
.
What is the kernel of this linear transformation.
A16)
Let 
be any two elements of 
Let a, b be any two elements of F.
We have
(
= 
So that f is a linear transformation.
To show that f is onto 
. Let 
be any elements 
.
Then 
and we have 
So that f is onto 
Therefore f is homomorphism of 
onto 
.
If W is the kernel of this homomorphism then 
We have
∀
Also if 
then 
Implies 
Therefore 
Hence W is the kernel of f.
Q17) 
be the linear operator defined by F(x, y) = (2x + 3y, 4x – 5y).
Find the matrix representation of F relative to the basis S = { 
} = {(1, 2), (2, 5)}
A17)
(1) First find F(
, and then write it as a linear combination of the basis vectors 
and 
. (For notational convenience, we use column vectors.) We have
And
X + 2y = 8
2x + 5y = -6
Solve the system to obtain x = 52, y =-22. Hence,
Now
X + 2y = 19
2x + 5y = -17
Solve the system to obtain x = 129, y =-55. Hence,
Now write the coordinates of 
and 
as columns to obtain the matrix
Unit – 5
Unit – 5
Unit – 5
Unit – 5
Unit – 5
Unit – 5
Linear Equations and Matrix Theory
Q1) What do you understand by a matrix?
A1)
A matrix is a rectangular arrangement of the numbers.
These numbers inside the matrix are known as elements of the matrix.
A matrix ‘A’ is expressed as-
The vertical elements are called columns and the horizontal elements are rows of the matrix.
The order of matrix A is m by n or (m× n)
Q2) What do you understand by the transpose of a matrix?
A2)
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 
Q3) What is the trace of a matrix?
A3)
Suppose A be a square matrix, then the sum of its diagonal elements is known as trace of the matrix.
Example- If we have a matrix A-
Then the trace of A = 0 + 2 + 4 = 6
Q4) Find the rank of a matrix M by echelon form.
M = 
A4)
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.
Q5) Find the rank of a matrix A by echelon form.
A = 
A5)
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.
Q6) What is characteristic equation?
A6)
Suppose we have A = 
be an n × n matrix , then
Characteristic equation- The equation |
= 0 is called the characteristic equation of A , where |
is called characteristic matrix of A. Here I is the identity matrix.
The determinant |
is called the characteristic polynomial of A.
Characteristic roots-the roots of the characteristic equation are known as characteristic roots or Eigen values or characteristic values.
Q7) Find the characteristic equation of the matrix A:
A = 
A7)
The characteristic equation will be-
|
= 0
= 0
On solving the determinant, we get
(4-
Or 
On solving we get,
Which is the characteristic equation of matrix A.
Q8) Find the characteristic equation and characteristic roots of the matrix A:
A = 
A8)
We know that the characteristic equation of the matrix A will be-
|
= 0
So that matrix A becomes,
= 0
Which gives , on solving
(1-
= 0
Or 
Or (
Which is the characteristic equation of matrix A.
The characteristic roots will be,
(
(
(
(
Values of 
are-
These are the characteristic roots of matrix A.
Q9) Find the characteristic equation and characteristic roots of the matrix A:
A = 
A9)
We know that, the characteristic equation is-
|
= 0
= 0
Which gives,
(1-
Characteristic roots are- 
Q10) What do you understand by Eigen values and Eigen vectors?
A10)
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.
Q11) Write down the properties of Eigen values.
A11)
Properties of Eigen Values:
- 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.
If 
are the Eigen values of the matrix A then 
has the Eigen values 
Q12) Find the sum and the product of the Eigen values of 
?
A12)
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 
Q13) Find out the Eigen values and Eigen vectors of 
?
A13)
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).
Q14) Find out the Eigen values and Eigen vectors of 
A14)
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).
Q15) Check whether the following system of linear equations is consistent of not.
2x + 6y = -11
6x + 20y – 6z = -3
6y – 18z = -1
A15)
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
Q16) The mapping 
defined by-
Is a linear transformation 
.
What is the kernel of this linear transformation.
A16)
Let 
be any two elements of 
Let a, b be any two elements of F.
We have
(
= 
So that f is a linear transformation.
To show that f is onto 
. Let 
be any elements 
.
Then 
and we have 
So that f is onto 
Therefore f is homomorphism of 
onto 
.
If W is the kernel of this homomorphism then 
We have
∀
Also if 
then 
Implies 
Therefore 
Hence W is the kernel of f.
Q17) 
be the linear operator defined by F(x, y) = (2x + 3y, 4x – 5y).
Find the matrix representation of F relative to the basis S = { 
} = {(1, 2), (2, 5)}
A17)
(1) First find F(
, and then write it as a linear combination of the basis vectors 
and 
. (For notational convenience, we use column vectors.) We have
And
X + 2y = 8
2x + 5y = -6
Solve the system to obtain x = 52, y =-22. Hence,
Now
X + 2y = 19
2x + 5y = -17
Solve the system to obtain x = 129, y =-55. Hence,
Now write the coordinates of 
and 
as columns to obtain the matrix
