applications of matrices and determinants one mark questions

Applications of matrices and determinants

ONE MARK QUESTIONS

PREPARED BY:R.RAJENDRAN. M.A., M. SC., M. ED.,K.C.SANKARALINGA NADAR HR. SEC. SCHOOL,CHENNAI-21

CHOOSE THE CORRECT ANSWER

844

422

211

0

4

0

2

1

1. The rank of the matrix is

2. The rank of the diagonal matrix is

(a) 1 (b) 2

(c) 3 (d) 4

(a) 1 (b) 2

(c) 3 (d) 4

Choose the correct answer

3. If A = [2 0 1], then the rank of AAT is

(a) 1 (b) 2

(c) 3 (d) 0

3

2

1

A If 4. , then the rank of AAT is

(a) 3 (b) 0

(c) 1 (d) 2

Choose the correct answer

5. If the rank of the matrix is 2, then is

a) 1 (b) 2

(c) 3 (d) any real number

01

10

01

6. If A is a scalar matrix with scalar k 0, of order 3, then A–1 is

(a) (b)

(c) (d) KI

Ik

12

Ik

13

Ik

1

Choose the correct answer

7. If the matrix has an inverse then the value of k

is

a) K is any real number (b) k = – 4

(c) k – 4 (d) k 4

541

31

231

k

8. If A = is then (adj A)A =

(a) (b)

(c) (d)

1/50

01/5

10

01

5-0

05

43

12

50

05

Choose the correct answer

9. If A is square matrix of order n then |adj A| is

(a) |A|2 (b) |A|n

(c) |A|n – 1 (d) |A|

10. If A is matrix of order 3 then det (kA) is

(a) k3 det (A) (b) k2 det (A)

(c) k det (A) (d) det (A)

Choose the correct answer

11. The inverse of the matrix is

(a) (b)

(c) (d)

100

010

001

001

010

100

100

010

001

001

010

100

001

010

100

Choose the correct answer

12. If I is the matrix of order n, where k 0 is a constant,

then adj (k I) is

(a) kn (adj I) (b) k (adj I)

(c) k2 (adj I) (d) k n–1(adj I)

13. If A and B are any two matrices such that AB = 0 and A

is non-singular, then

(a) B = 0 (b) B is singular

(c) B is non-singular (d) B = A

Choose the correct answer

14. If A = then A12 =

(a) (b)

(c) (d)

600

00

1250

00

00

00

50

00

10

01

15. The inverse of

(a) (b)

(c) (d)

35-

1-2

3-1

52-

3-5-

1-3

25

13

2-1

53-

Choose the correct answer

16. In a system of 3 linear non-homogeneous equation with

three unknowns, if = 0 and x = 0, y 0 and z = 0 then

the system has

(a) unique solution (b) two solutions

(c) infinitely many solutions (d) no solutions

Choose the correct answer

17. The system of equations ax + y + z = 0; x + by + z = 0;

x + y + cz = 0 has a non trivial solution then

(a) 1 (b) 2

(c) – 1 (d) 0

cba 1

1

1

1

1

1

Choose the correct answer

18. If aex + bey = c; pex + qey = d and 1 =

2 = 3 = then the value of (x, y) is

qp

ba

qd

bc

dp

ca

1

3

1

2 , )(a

1

3

1

2 log,log )(b

2

1

3

1 log,log )(c

3

1

2

1 log,log )(b

Choose the correct answer

19. If the equations –2x + y + z = l; x – 2y + z = m;

x + y – 2z = n such that l + m + n = 0, then the system has

(a) a non-zero unique solution

(b) trivial solution

(c) infinitely many solution

(d) no solution

Choose the correct answer

20. Given (A, B) = (A) = number of unknowns, then the

system has

(a) unique solution

(b) no solution

(c) inconsistent

(d) infinitely many solution

Choose the correct answer

21. Given (A, B) = (A) < number of unknowns, then the

system has

(a) unique solution

(b) no solution

(c) 3 solutions

(d) infinitely many solution

Choose the correct answer

22. Given (A, B) (A), then the system has

(a) unique solution

(b) no solution

(c) 3 solutions

(d) infinitely many solution

Choose the correct answer

23. Which one of the following is not true regarding the

non-singular matrices A, B, C?

(a) A(adjA) = (adj A) A

(b) (AB)–1 = B–1 A–1

(c) (AB)T = BT AT

(d) (AT)–1 (A–1)T

Choose the correct answer

24. The value of

(a) 0 (b) abc

(c) a + b + c (d) – abc

c

b

a

00

00

00

Choose the correct answer

25. The co-factor of – 1 in

(a) 38 (b) 48

(c) 28 (d) – 38

242

531

753

Choose the correct answer

26. If the equation – 2x + y + z = l; x – 2y + z = m;

x + y – 2z = n such that l + m + n = 0, then the system

has

(a) a non-zero unique solution

(b) trivial solution

(c) infinitely many solution

(d) no solution

Choose the correct answer

27. The rank of the matrix is

(a) 1 (b) 2

(c) 0 (d) 8

28. The rank of the matrix is

(a) 9 (b) 2

(c) 1 (d) 5

21

42

12

17

Choose the correct answer

29. If A and B are matrices conformable to multiplication then (AB)T is

(a) ATBT (b) BTAT

(c) AB (d) BA

30. (AT)– 1 is equal to

(a) A– 1 (b) AT

(c) A (d) (A– 1)T

Choose the correct answer

31. If (A) = r then which of the following is correct?

(a) all the minors of order r which do not vanish

(b) A has atleast one minor of r which does not vanish

(c) A has atleast one (r + 1) order minor which vanishes

(d) all (r + 1) and higher order minors should not

vanish

Choose the correct answer

32. Which of the following is not elementary

transformation?

(a) RiRj (b) Ri 2Ri + Rj

(c) Ci Ci + Cj (d) Ri Ri + Cj

33. Cramer’s rule is applicable only (with three unknowns)

when

(a) 0 (b) = 0

(c) = 0, x 0 (d) = x = y = z = 0

Choose the correct answer

34. Equivalent matrices are obtained by

(a) taking inverses

(b) taking transposes

(c) taking adjoints

(d) taking finite number of elementary transformations

Choose the correct answer

35. In echelon form, which of the following is incorrect?

(a) Every row of A which has all its entries 0 occurs

below every row, which has a non-zero entry.

(b) The first non-zero entry in each non-zero row is 1

(c) The number of zeroes before the first non-zero

element in a row is less than the number of such

zeroes in the next row

(d) Two rows can have same number of zeroes before

the first non-zero entry

Choose the correct answer

36. If 0 then the system is

(a) consistent and has unique solution

(b) consistent and has infinitely many solution

(c) inconsistent

(d) Either consistent or inconsistent

Choose the correct answer

37. In the system of 3 linear equations with three

unknowns, if = 0 and one of x, y or z is non-zero

then the system is

(a) consistent (b) inconsistent

(c) consistent and the system reduces to two equations

(d) consistent and the system reduces to a single

equation

Choose the correct answer

38. In the system of 3 linear equations with three

unknowns, if = 0 and all 2 2 minors of = 0 and

atleast one 2 2 minor of x or y or z is non-zero then

the system is

(a) consistent

(b) inconsistent

(c) consistent and the system reduces to two equations

(d) consistent and the system reduces to a single

equation

Choose the correct answer

39. In the system of 3 linear equations with three

unknowns, if = 0, x = 0, y = 0, z = 0 and atleast one

2 2 minor of 0 then the system is

(a) consistent

(b) inconsistent

(c) consistent and the system reduces to two equations

(d) consistent and the system reduces to a single

equation

40. In the system of 3 linear equations with three unknowns,

if = 0 and all 2 2 minors of , x, y, z are zeros and

atleast one non-zero element is in then the system is

(a) consistent (b) inconsistent

(c) consistent and the system reduces to two equations

(d) consistent and the system reduces to a single

equation    

41. Every homogeneous system

(a) is always consistent

(b) has only trivial solution

(c) has infinitely many solution

(d) need not be consistent

42. If (A) = (A,B) then the system is

(a) consistent and has infinitely many solution

(b) consistent

(c) consistent and has a unique solution

(d) inconsistent

43. If (A) = (A,B) = the number of unknowns then the

system is

(a) consistent and has infinitely many solution

(b) consistent

(c) consistent and has a unique solution

(d) inconsistent

Choose the correct answer

44. In the system of 3 linear equations with three

unknowns if (A) = (A,B) = 1 then the system is

(a) has a unique solution

(b) reduces to 2 equations and has infinitely many

solutions

(c) reduces to a single equation and has infinitely

many solution

(d) inconsistent           

45. In the homogeneous system with three unknowns if

(A) = number of unknowns then the system has

(a) only trivial solution

(b) reduces to 2 equations and has infinitely many

solutions

(c) reduces to a single equation and has infinitely

many solution

(d) inconsistent 

46. In the system of 3 linear equations with three unknowns,

in the non-homogeneous system if (A) = (A,B) = 2

then the system has

(a) has a unique solution

(b) reduces to 2 equations and has infinitely many

solutions

(c) reduces to a single equation and has infinitely many

solution

(d) inconsistent

Choose the correct answer

47. In the homogeneous system with three unknowns if

(A) < number of unknowns then the system has

(a) only trivial solution

(b) trivial solution and infinitely many non-trivial

solutions

(c) only non-trivial solutions

(d) no solution

Choose the correct answer

48. Which of the following statement is correct regarding

homogeneous system?

(a) always inconsistent

(b) has only trivial solution

(c) has only non trivial solutions

(d) has trivial solution only if rank of the coefficient

matrix is equal to the number of unknowns