1.Explain transpose of a matrix with an example.

2.Show that any square matrix can be expressed as the sum of a symmetric and a skew symmetric matrix.

3.If A and B are non-singular matrices of same order, then show that

(AB)^-1 = B^-1A^-1

4. Find the inverse of the matrix

A = [

0	1	1
1	0	1
1	1	0

]

5. Solve the following equations by matrix method

x + y + z = 6
x − y + z = 2
2x + y − z = 1

6. Show that

(3x + 4y + 5) dx + (4x − 3y + 3) dy = 0

is an exact equation and hence solve it.

7. Solve the differential equation :

d²y dx²  − 3  dy dx  + 2y = e^2x

8. Find A + B, if

A = [

2	2	1
1	5	0
0	0	1

]
  and  
B = [

5	7	1
0	3	0
1	0	8

]

9. If A and B are symmetric matrices, then show that AB is symmetric if and only if A and B commute.

10. Show that every square matrix can be expressed as the sum of symmetric and skew-symmetric matrices.

11. Solve the following differential equation :

dy dx + ay + b = 0,  a ≠ 0

12. What are Hermitian matrices? Show that the eigenvalues of Hermitian matrix are real.

13. Find the inverse of the matrix

A = [

1	3
2	1

]

14. Solve the following by the method of integrating factor :

x dy dx + y = x³ + x

15.Solve the following differential equation by matrix method :

x+5y+3z=1
3x+y+2z=1
x+2y+z=0

16. Find the values of x, y and z which satisfy the matrix equation

[

x + 3	2y + x
z − 1	4a − 6

]  =  [

0	−7
3	2a

]

17. If

A = [

0	1	2
1	2	3
2	3	4

]  and  B = [

1	−2
−1	0
2	−1

]

then obtain the product AB.

18. What do you mean by ‘order’ and ‘degree’ of a differential equation?

19. If

A = [

1	2	2
2	1	2
2	2	1

]

then show that

A² − 4A − 5I = 0

where I and O are unit matrix and null matrix of order 3 respectively.

20. If

A = ¹/₉ [

−8	1	4
4	4	7
1	−8	4

]

then prove that

A⁻¹ = A^'

A^' being the transpose of A.

21. Solve the following differential equation by the method of integrating factor

(x³ − x) dy dx − (3x² − 1)y = x⁵ − 2x³ + x

22. Solve the differential equation

(2xy + x²)dy = (3y² + 2xy)dx

23. Which of the following obey commutative law?

A + B,    A − B,    A · B,    A × B

24.Show that vector product of two vectors is a vector.

25. Define singular and non-singular matrices.

26.Prove that

a × (b × c) + b × (c × a) + c × (a × b) = 0

27. Prove that

i × (a × i) + j × (a × j) + k × (a × k) = 2a

where a is a vector.

28. Show that any square matrix can be uniquely expressed as the sum of symmetric matrix and antisymmetric matrix.

29. Find the inverse of the matrix

[

1	−1	3
−1	1	2
3	2	−1

]

30. Define dot and cross products of two vectors.

31.. What are meant by symmetric and skew-symmetric matrices?

32. Show that the vectors

A = 2i − 3j − k   and   B = −6i + 9j + 3k

are parallel.

33. Define vector triple product. Show that

A × (B × C) = (A · C) B − (A · B) C

34. Show that the vectors

A = 3i − 2j + k,

B = i − 3j + 5k and C = 2i + j − 4k

form a right-angled triangle.

35. Define Hermitian and skew-Hermitian matrices. Show that every square matrix can be uniquely expressed as the sum of Hermitian and skew-Hermitian matrices.

36. What is diagonalization of a matrix? Diagonalize the matrix

A = [

4/3	√2/3
√2/3	5/3

]