MATHEMATICS
Matrices
Practice questions
I
ANSWER ALL THE QUESTIONS:
1 Define a scalar matrix.
2. Define a diagonal matrix.
3. Define an Identity matrix.
 1
4. If A  4 , B  -1 2 1 , Verify that AB  BA
 3 
5. Construct a 2  2 matrix A  a ij  whose elements are given by a ij 
6. Construct a 22 matrix, A= [aij], where elements are given by, a ij 
7. If A = [aij], where elements are given by aij 
1 MARK
1
3i  j
2
i
.
j
1
3i  j, construct 22 matrix.
2
 x  2 y  3
8. If 
 is a scalar matrix, find x and y.
 0
4 
9. If a matrix has 5 elements, what are the possible orders it can have?
II
ANSWER ALL QUESTIONS:
3 MARKS
3 5 
1. Express A  
 as sum of symmetric and skew symmetric matrix.
1 1
 1 5
2. Express A  
 as sum of symmetric and skew symmetric matrices.
1 2
3. For the square matrix with real number entries show that
(i) A + A’ is
s a symmetric matrix. (ii) A – A’ is a skew-symmetric matrix,
trix,
1 5
4. For the matrix A    verify that
6 7
(i) A + A’ is
s a symmetric matrix. (ii) A – A’ is a skew-symmetric matrix,
trix,
5. If A and B are symmetric matrices
ma
of the same order, then show that AB is
symmetric if and only if A and B commute that is AB = BA
BA.
III
1.
ANSWER ALL QUESTIONS:
1 2
2 0
1 1
If A  
, B 
 and C  
 , calculate AC, BC and (A + B)C.
2 1
1 3
2 3
Also
lso verify that (A + B ) C = AC + BC
5 MARKS
2. For any square matrix A with real numbers, prove that
A + A’ is a symmetric and
A – A’ is skew – symmetric
 1 2 3


3. If A  3 2 1 , then show that A3 – 23A – 40I = 0


 4 2 1
1 2

4. If A  5 0

1 1
Also verify A+
3
3 1 2
4 1 2





2  , B  4 2 5 and C  0 3 2 then compute (A + B) and (B – C)





 2 0 3
1 2 3
1 
(B – C) = (A + B) – C.
 2
 
5. If A   4  , B  1 3 6 , verify that (AB)| = B|A|
 
 5 
 0 6 7
0 1


6. If A  6 0 8, B  1 0
 7 8 0
1 2
that (A+B) C = AC + BC.
1
2

2 and C  2 , calculate AC, BC and (A+B) C. Also, verify
 3 
0
1 0 2


7. If A  0 2 1 prove that A3 – 6A2+7A+2I=0.
2 0 3
1 2
2 0
1
,B  
and C  
8. If A 


2 1
1 3
2
AB+AC = A(B+C).
9.
1
, Calculate AB, AC and A(B+C). verify that
3
1 2 3
If A  3 2 1 then show that A3 – 23A – 40l = 0.


4 2 1
1 2 −3
3 −1 2
4 1 2
10. If 𝐴 = 5 0
2 , 𝐵 = 4 2 5 𝑎𝑛𝑑 𝐶 = 0 3 2
1 −1 1
2 0 3
1 −2 3
(𝐵 − 𝐶 ). Also verify that 𝐴 + (𝐵 − 𝐶 ) = (𝐴 + 𝐵 ) − 𝐶.
cos 𝑥
11. If 𝐹 (𝑥 ) = sin 𝑥
0
PYQP-2014-2019
– sin 𝑥
cos 𝑥
0
then compute (𝐴 + 𝐵 ) and
0
0 than show that 𝐹 (𝑥)𝐹 (𝑦) = 𝐹 (𝑥 + 𝑦).
1
**************
Page 2