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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
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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
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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
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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
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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
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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)
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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
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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
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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-
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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
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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
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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
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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
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20. Given (A, B) = (A) = number of unknowns, then the
system has
(a) unique solution
(b) no solution
(c) inconsistent
(d) infinitely many solution
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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
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22. Given (A, B) (A), then the system has
(a) unique solution
(b) no solution
(c) 3 solutions
(d) infinitely many solution
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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
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24. The value of
(a) 0 (b) abc
(c) a + b + c (d) – abc
c
b
a
00
00
00
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25. The co-factor of – 1 in
(a) 38 (b) 48
(c) 28 (d) – 38
242
531
753
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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
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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
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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
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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
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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
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34. Equivalent matrices are obtained by
(a) taking inverses
(b) taking transposes
(c) taking adjoints
(d) taking finite number of elementary transformations
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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
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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
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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
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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
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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
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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
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41. Every homogeneous system
(a) is always consistent
(b) has only trivial solution
(c) has infinitely many solution
(d) need not be consistent
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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
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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
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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
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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
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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
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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
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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
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