MATRICES
1. Employing elementary transformations, find the inverse of the matrix
 3  3 4


(i) 2  3 4


0  1 1
0 1 2 


(ii) 1 2 3 (iii)


3 1 1
 1 1 0 


Ans: (i)  2 3  4


 2 3  3
1
2  3 4 

2  3 4 (iv)  1


2
0  1 1

1
2 3 1
3 3 2

4 3 3

1 1 1
 1 1 1 
1
(ii)
 8 6  2  (iii) Does not exist (iv)

2
 5  3 1 
0 
 1 2 1
 1  2 2  3


0
1 1 1 


 2 3  2 3 
2. Find the rank of the following matrices by reducing to normal form:
1 2 3 2


(i) 2 3 5 1 (ii)


 1 3 4 5
Ans:
(i) 2
2 1 4 
1
0 0
2 1  3  6 
2

0 4
4
3
4
3  3 1



(iv) 
2  (iii)

1
0 9
2
3
4
 1 1
1
2 



 1  2 6  7 
0 10
(ii)3
(iii) 3
0
0
0
5 0 0

1 1 2 

0 1 11
(iv) 3
2
1 1

3. For the matrix A = 1 2
3  , find non singular matrices P and Q such that PAQ is in normal form.

0  1  1
 3  3 4


4. If A  2  3 4 ,find two non singular matrices P and Q such that PAQ=I. Hence find A 1


0  1 1
 1 1 0 


Ans:  2 3  4


 2 3  3
5.
Find the rank following matrices by reducing them to echelon (triangular ) form:
1 2 3 2


(i) 2 3 5 1


 1 3 4 5
Ans (i) 2
1 2  1
4 1
2
(ii) 
3  1 1

0
1 2
(ii) 3
3
1

2

1
1
2
(iii) 
1

0
3
2
5
1
2 1 6 3 

1 2 3  1

2 5 2  3
(iii) 3
 3 P P


6. Find the value of P for which the matrix A= P 3 P is be of rank 1. Ans: P=3


 P P 3 
 1 1  1 0
 4 4  3 1
 Ans:   6
7. Determine the values of  such that the rank of A is 3, where A= 
 2 2 2


 9 9  3
1. Solve with the help of Gauss elimination method:
2 x 1 + x 2 + 2 x 3 + x 4 =6, 6 x 1 -6 x 2 + 6 x 3 + 12 x 4 =36, 4 x 1 + 3 x 2 + 3 x 3 -3 x 4 =-1, 2 x 1 + 2 x 2 - x 3 + x 4 =10
Ans: x 1 =2, x 2 =1 x 3 =-1, x4=3.
2. Test the consistency of following system of linear equations and hence find the solution:
-1-
4 x 1 - x 2 =12, - x 1 +5 x 2 -2 x 3 =0, -2 x 2 +4 x 3 =-8
Ans: Unique solution, x 1 =
44
4
 32
, x2 =
, x3 =
15
15
15
3. Show that the system of equations x + y + z=-3, 3x+y-2z=-2 and 2x+4y+7z=7 is not consistent.
Verify that the following system of equations is inconsistent:
x + 2y +2 z=1,
2x+y+z=-2
3x+2y+2z=3 and
y + z=0
4. Test the consistency of following system of equations:
5x +3y +7z=4,
3x+26y+2z=9
7 x +2y+11z=5.
Ans: Consistent
5. Show that the system of equations :3x+4y+5z=A,
4x+5y+6z=B, 5x+6y+7z=C are consistent only if
A,B and C are in arithmetic progression.
6. For what value of k, the equations x + y + z=1, 2x+y+4z=k and 4x+y+10z= k 2 have a solution and solve
them completely in each case.
Ans: k=1,k=2.When k=1;x= -3 k 1 ,y=2 k 1 +1,z= k 1 , When k=2;x= 1-3 k 2 ,y=2 k 2 ,z= k 2 .
7. Investigate for what values of λ and μ do the system of equations
2x -5 y +2 z=8, 2x+4y+6z=5, x+2y+ λ z= μ have (i) No solution (ii) Unique solution (iii) Infinite solutions?
Ans: (i) λ=3, μ≠
5
5
(ii) λ≠3 (iii) λ=3, μ= .
2
2
 3  2 1  x   b 

   
8. Determine the values of a and b for which the system 5  8 9 y  3 has (i) a unique solution (ii)

   
2 1 a  z   1
1
1
no solution (iii) infinitely many solutions. Ans: (i) a≠-3 (ii) a  3 , b  (iii) a  3 , b  .
3
3
9. Show that the system of equations 3x  4y  5z  a , 4x  5y  6z  b , 5x  6y  7z  c does not have a
solution unless a+c=2b.
10. Show that the system of equations x + 2y – 2u=0, 2x-y-u=0, x+2z-u=0 and 4x - y + 3z-u=0 do not have a
non trivial solution. Ans:x=0,y=0,z=0,u=0
11. Discuss consistency and hence solve: x+3y-2z=0, 2x-y+4z=0, x-11y+14z=0 Ans:x=-10k,y=8k,z=7k
 1 2 1 


12. If A= 3  1 2 , find the values of  for which the matrix equation AX=O has (i) Unique solution (ii)


 0
1  
More than one solution.
Ans: (i)  ≠1 (ii)  =1.
13. Show that the equations -2x+y+z=a, x-2y+z=b and x+y-2z=c have no solution unless a+ b + c=0, in which
case they have infinitely many solutions. Find these solutions when a=1, b=1, c=-2.
Ans: x=k-1, y=k-1and z=k.
14. Find the values of k for which the system of equations (3k-8) x+3y+3z=0, 3x + (3k-8) y+3z=0,
3x+3y+ (3k-8) z=0 has a non trivial solution.
Ans: k=
2 11 11
,
,
.
3 3 3
15. Find the values of  for which the following system of equations is consistent and has non trivial
solutions.Solve equations for all such values of  :
  1x  3  1y  2z  0 ,   1x  4  2y    3z  0 2x  3  1y  3  1z  0
Ans:   0 ,3 ;For   0, x  y  z  k 1 ; For   3, x  5k 3  3k 2 , y  k 3 , z  k 2
16. Show that the homogeneous system of equations x+ y cos  +z cos  =0, x cos  +y +z cos  =0, x cos  +y
cos  +z=0 has non trivial solution if  +  +  =0.
1. Find whether or not the following set of vectors is linearly dependent or independent:
[1, 1, 1, 1], [0, 1, 1, 1], [0, 0, 1, 1], [0, 0, 0, 1].
Ans: Linearly independent.
2. If X 1 =[3,1,-4], X 2 =[2,2,-3] and X 3 =[0,-4,1],then show that:
(i) The vectors X 1 and X 2 are linearly independent.
(ii) The vectors X 1 , X 2 and X 3 are linearly dependent.
-2-
3. Show that the row vectors of
4.
5.
6.
7.
1
the matrix  1

0
2
3
2
2

0  are linearly independent.
1

1 0 0


Show that the column vectors of A= 6 2 1 are linearly independent.


 4 3 2 
Show that the vectors x 1 =(1,2,4), x 2 =(2,-1,3), x 3 =(0,1,2)and x 4 =(-3,7,2) are linearly dependent and find the
relation between them.
Ans: 9 x 1 -12 x 2 + 5 x 3 - 5 x 4 =0.
Show that the vectors X 1 =[2,3,1,-1], X 2 =[2,3,1,-2] , X 3 =[4,6,2,1] are linearly dependent. Express one of the
vectors as a linear combination of others.Ans: 5 X 1 -3 X 2 = X 3
Find the value of  for which the vectors (1,2,  ), (2,1,5) and (3,5,7 ) are linearly dependent.
5

14
Show that the vectors X 1 = [ a1 , b1 ] and X 2 = [ a 2 , b 2 ] are linearly dependent if and only if a 1 b2 - a 2 b1 =0.
8.
1. Find the eigen values and corresponding eigen vectors of the following matrices
3
(i) 0
0
1 4
1   1  3
1   2 
 8 6 2 
     
   


2 6  (ii)  6 7  4 Ans: (i)3,2,5  0  ,  1 ,  2  (ii)0,3,15  2  ,  1 
 0   0   1 
 2   2 
0 5
  2  4 3 
2 1 1 
2
 
 2
 1 
2. Prove that for the matrix A=  2
3 4  , all its eigen values are distinct and real. Hence find corresponding
1 1  2
1 0  2 
     
vectors. Ans: λ =1,-1,3;  1 ,  1 , 3
     
 0   1    1
3. Show that the matrix A has repeated eigen values. Also find the corresponding eigen vectors, where
0 2
2
1  1 


   
A=  1 3 1 Ans:  =2, 2, 4; 1 , 0


   
 1  1 3
0    1
 3 10 5 


4. Show that the matrix  2  3  4 has less than 3 linearly independent eigen vectors. Also find them.


 3
5
7 
1
5
 
 
Ans:  =2, 2, 3.For  =3, X1=  1  , For  =2, X2=  2 
 2 
 5
 6 2 2 


5. Find the eigen values and corresponding eigen vectors of the matrix   2 3 1
 2 1 3 
1 
1  2 
 
   
Ans: 2,2,8 k1  2  + k 2  0  ,  1
 0 
 2   1 
 5 2 
 .
6. Find the eigen values & corresponding eigen vectors of the matrix A  
 2  2
-3-
 1   2

Ans:-1,-6,  , 
2
1
  
2 1  1


7. Find the sum and product of the eigen values of the matrix A= 3 4 2  Ans:8, 8
1 0 2 
4 3 1 
 5  1 7 
1 


7. Using Cayley Hamilton theorem, find the inverse of 2 1  2 .Ans:
 4 3 10 


11 
 1 2 1 
 3  5  2 
 2 1 1 


8. Verify Cayley Hamilton Theorem for the matrix A=  1 2  1 .Hence compute A-1.


 1  1 2 
 3 1 1
1
Ans:  1 3 1 
4
1 1 3 
9.
1
Given A = 0
3
2
1
1
1
0
1 , find adjA by Cayley Hamilton Theorem. Ans:  3
3
1

1 1
4
7
1 .
1

2 1 1 


10. Find the characteristic equation of the matrix A= 0 1 0 and hence compute A 1 .Also find the matrix


 1 1 2 
represented by A 8 -5 A 7 +7 A 6 -3 A 5 + A 4 -5 A 3 +8 A 2 -2A+I.
 2 1 1 8 5 5
3
2
1 1
Ans:  -5  +7  -3=0, A =  0 3 0  , 0 3 0 
3
1 1 2  5 5 8 
1 0 0 


11. If A  1 0 1 ,show that for every integer n  3, A n  A n 2  A 2  I


0 1 0
12. State and prove Cayley Hamilton theorem.
5  2 i  3
 3

1. If H= 5  2i
7
4i  , show that H is a Hermitian matrix. Verify that iH is a skew Hermitian matrix.

  3
 4i
5 
2  i 3  1  3i 
2. If A= 
, verify that A  A is a Hermitian matrix where A  is the conjugate transpose of A.

  5 i 4  2i 
1  1 1 i
3. Show that the matrix

 is unitary.
3 1  i  1 
 8 4 1 
1
4. Prove that the following matrix is orthogonal:  1 4  8
9
 4 7 4 
  i    i 
2
2
2
2
5. Show that the matrix 
is unitary if and only if        =1.

   i   i 
-4-
0
1  2i 
, obtain the matrix I N  I  N 1 and show that it is unitary.

0 
 1  2 i
2  3i 4  5i 
 i

7. Express the matrix A  6  i
0
4  5i  as a sum of a Hermitian and skew Hermitian matrix.

  i
2  i 2  i 
4  2i 2  3i   i
 2  i 2  2i 
 0



Ans: 4  2i
0
3  2i   2  i
0
1  3i 

 2  3i 3  2i
2   2  2i  1  3i
i 

6. If N= 
 1 i

8. Verify that the matrix A=  2
1i

 2
1i 
2  has Eigen values with unit modulus.
1 i 

2 
9. If A is any square matrix, prove that A+A*, A A*,A*A are Hermitian and A-A* is skew Hermitian.
10. If  be an eigen value of a non singular matrix A, show that
(i) 1 is an eigen value of A-1.
(ii)
A

is an eigen value of adjA.
11. Prove that every eigen vector corresponds to a unique eigen value.
12. Prove the following:
a) Latent roots of a Hermitian matrix are all real.
b) The characteristic root of a skew Hermitian matrix is either zero or a purely imaginary number.
c) The characteristic roots of a unitary matrix are of unit modulus.
13. Show that the product of eigen values of a square matrix A is equal to detA.
14. Show that the sum of eigen values of a square matrix is equal to the sum of the elements of its principal
diagonal.
 4 1
1. Find a matrix P which diagonalizes the matrix A  
.Verify that P 1AP =D,where D is the

 2 3
 1 1
2 0
diagonal matrix .Ans: P  
,D= 


 2 1
 0 5
2  2
1
2. A square matrix is defined by A   1
2
1  .Find
 1  1 0 
 2  2 2 
1 0
diagonal matrix D of A. Ans: P   1
1
1  ,D= 0  1
 1  1  1
0 0
1 2  2
the modal matrix P and resulting
0
0
3

5
0 
 1 2 1  to diagonal form.
1 1 0 
0  5

1 1 2 
4. Reduce the matrix A= 0 2 1 to diagonal form by similarity transformation. Hence find A 3 .
0 0 3 
1 0 0 
1  7 32 
3
Ans: D = 0 2 0 , A = 0 8 19 
0 0 3
0 0 27 
3. Reduce the matrix A=
-5-
1
Ans: 0
0
0
0
 3 1  1


5. Show that the matrix A=  2 1 2  is diagonalizable.
 0 1 2 
 1 2 2


6. Show that the matrix A=  0 2 1 is not diagonalizable.
 1 2 2
-6-