50 questions on linear algebra for NET and GATE aspirants

Find the correct options:

 

1)  M=\left(\begin{array}{ccc}1 & 2 & 2 \\0 & 2 & 2 \\0 & 1 & 1 \end{array}\right) and V={ Mx^{T} : x\in R^{3}}. Then dim V is

(a)   0   (b)   1   (c)   2   (d)   3

 

2)  A^{2}-A=0, where A is a 9\times 9 matrix. Then

(a) A must be a zero matrix       (b) A is an identity matrix

(c) rank of A is 1 or 0                   (d) A is diagonalizable

 

3) The number of linearly independent eigen vectors of \left(\begin{array}{cccc}1 & 1 & 0 & 0 \\2 & 2 & 0 & 0 \\0 & 0 & 3 & 0\\0 & 0 & 5 & 5 \end{array}\right)  is

(a)   1  (b)  2  (c)   3  (d)   4

 

4) The minimal polynomial of \left(\begin{array}{cccc}2 & 1 & 0 & 0 \\0 & 2 & 0 & 0 \\0 & 0 & 2 & 0\\0 & 0 & 0 & 5 \end{array}\right) is

(a) (x-2)   (b)  (x-2)(x-5)  (c)  (x-2)^{2}(x-5)  (d)  (x-2)^{3}(x-5)

 

5) A is a unitary matrix. Then eigen value of A are

(a)  1, -1  (b)  1, -i  (c)  i, -i   (d)   -1, i

 

6) \left(\begin{array}{ccc}2 & -3 \\2 & -2 \end{array}\right) is an operator on R^{2}. The invariant subspaces of the operator are

(a) R^{2} and the subspace with base {(0,1)}    (b) R^2 and the zero subspace

(c) R^2, the zero subspace and the subspace with base {(1,1)}     (d) only R^{2}

 

7) Rank of the matrix \left(\begin{array}{ccccc}21 & -7 & 0 & 0 & 0 \\-11 & 9 & 0 & 0 & 0 \\0 & -19 & 35 & 0 & 0 \\0 & 15 & 0 & 12 & 20 \\0 & 0 & -24 & 21 & 35 \end{array}\right)  is

(a)  2  (b)  3  (c)  4   (d)   5

 

8) The dimension of the subspace of M_{2\times 2} spanned by \left(\begin{array}{ccc}1 & -5 \\-4 & 2 \end{array}\right),  \left(\begin{array}{ccc}1 & 1 \\-1 & 5 \end{array}\right) and \left(\begin{array}{ccc}2 & -4 \\-5 & 7 \end{array}\right) is

(a)  1  (b)  2  (c)  3   (d)   4

 

9) U and V are subspace of R^{4} such that

U = span [(1,2,3,4), (5,7,2,1), (3,1,4,-3)]

V=span [(2,1,2,3), (3,0,1,2), (1,1,5,3)].

Then the dimension of U\cap V is

(a)  1  (b)  2  (c)  3   (d)   4

 

10)  Let M_{n\times n} be the set of all n-square symmetric matrices and the characteristics polynomial of each A\in M_{n\times n} is of the form

t^{n}+t^{n-2}+a_{n-3}t^{n-3}+\dots +a_{1}t+a_{0}. Then the dimension of M_{n\times n} over R is

(a)  \frac{(n-1)n}{2}  (b)  \frac{(n-2)n}{2}  (c)  \frac{(n-1)(n+2)}{2}   (d)   \frac{(n-1)^{2}}{2}

 

11) A is a 3\times 3 matrix with \sigma (A)={1, -1, 0 }. Then |I+A^{100}| is

(a)  6  (b)  4  (c)   9  (d)   100

 

12) A is 5\times 5 matrix, all of whose entries are 1, then

(a)  A is not diagonalizable  (b)  A is idempotent  (c)   A is nilpotent

(d)   The minimal polynomial and the characteristics polynomial of A are not equal.

 

13) A is an upper triangular with all diagonal entries zero, then I+A is

(a)  invertible  (b)  idempotent  (c)  singular   (d)   nilpotent

 

14) Number of linearly independent eigen vectors of \left(\begin{array}{cccc}2 & 2 & 0 & 0 \\2 & 1 & 0 & 0 \\0 & 0 & 3 & 0 \\0 & 0 & 1 & 4 \end{array}\right) is

READ:   Linear Algebra Questions for NET/GATE

(a)  1  (b)  2  (c)  3   (d)   4

 

15) A is a 5\times 5 matrix over R, then (t^{2}+1)(t^{2}+2)

(a)  is a minimal polynomial  (b)  is a characteristics polynomial

(c)  both (a) and (b) are true  (d)   none of (a) and (b) is true

 

16) M is a 2-square matrix of rank 1, then M is

(a)  diagonalizable and non singular  (b)  diagonalizable and nilpotent

(c)  neither diagonalizable nor nilpotent   (d)   either diagonalizable or nilpotent

 

17)  A be a n-square matrix with integer entries and B=A+\frac{1}{2} I. Then

(a)  B is idempotent  (b)  B^{-1} exist  (c)  B is nilpotent   (d)   B-I is idempotent

 

18) Let A\in M_{3\times 3}(R), then t^{2}+1 is

(a)  a minimal polynomial of A  (b)  a characteristics polynomial of A

(c)  both (a) and (b) are true   (d)   none of (a) and (b) is true

 

19)  A is a 4-square matrix and A^{5}=0. Then

(a)  A^{4}=I  (b)   A^{4}=A   (c)   A^{4}=0   (d)   A^{4}=-I

 

20) A=\left(\begin{array}{ccc}0 & 1 & a \\0 & 0 & 1 \\0 & 0 & 0 \end{array}\right) and B=\left(\begin{array}{ccc}0 & 1 & 0 \\0 & 0 & 1 \\0 & 0 & 0 \end{array}\right).  Then

(a)  A and B are similar  (b)  A and B are not similar

(c)  A and B are nilpotent   (d)  A and AB are similar

 

21) Let S={ 2-x+3x^{2}, x+x^{2}, 1-2x^{2} } be subset of P_{2}(R). Then

(a)  S is linearly independent  (b)  S is linearly dependent

(c)   (2,-1,3), (0,1,1), (1,0,-2) are linearly dependent  (d)  S is a basis of P_{2}(R)

 

22) T: P_{2}(R)\rightarrow P_{3}(R) such that T(f(x))=2f^{ '}(x)+3 \int_{0}^{x}f(t)dt. Then rank of T is

(a)  1  (b)  2  (c)  3   (d)   4

 

23) T_{i}: P(R)\rightarrow P(R) such that T_{1}(f(x))= \int_{0}^{x}f(t)dt and T_{2}(f(x))=f^{'}(x). Then

(a) T_{1} is 1-1, T_{2} is not  (b)  T_{2} is 1-1, T_{1} is not

(c)  T_{1} is onto and T_{2} is 1-1   (d)   T_{1} and T_{2} both are 1-1

 

24) T: P_{3}(R)\rightarrow M_{2\times 2}(R), such that T(f(x))= \left(\begin{array}{ccc}f(1) & f(2)\\f(3) & f(4)\end{array}\right), then

(a)  T is 1-1  (b)  T is onto  (c)  T is both 1-1 and onto   (d)  T is neither 1-1 nor onto

 

25) W=L{(1,0,0,0), (0,1,0,0)}, then

(a)  \frac{R^{4}}{W}=L{ W+(2,0,0,0), W+(0,2,0,0) }

(b)  \frac{R^{4}}{W}=L{ W+(1,2,3,4), W+(2,3,4,5) }

(c)  \frac{R^{4}}{W}=L{ W+(0,0,2,0), W+(0,0,0,2) }

(d)   { W+(1,2,3,4), W+(2,3,4,5) } is a basis of \frac{R^{4}}{W}

 

26) A= \left(\begin{array}{ccc}0 & 1 \\0 & 0 \end{array}\right), then

(a)  A has zero image  (b)  all the eigen value of A are zero

(c)  A is idempotent   (d)   A is nilpotent

 

27) T:R^{4}\rightarrow R^{4}, defined by T(e_{1})=e_{2}, T(e_{2})=e_{3}, T(e_{3})=0, T(e_{4})=e_{3}. Then

(a)  T is nilpotent  (b)  T has at least one non-zero eigen value

(c)  index of nilpotent is three   (d)   T is not nilpotent

READ:   H.C.F. and L.C.M.

 

28) A= \left(\begin{array}{ccc}a & a & a \\a & a & a \\a & a & a \end{array}\right), where a\neq 0, then

(a)  A is not diagonalizable  (b)  A is idempotent

(c)  A is nilpotent   (d)   minimal polynomial ≠ characteristics polynomial

 

29) A\in M_{2\times 2}(R) and rank of A is 1, then

(a)  A is diagonalizable  (b)  A is nilpotent

(c)  both (a) and (b) are true   (d)   none of (a) and (b) is true

 

30) A is a 3-square matrix and the eigen values of A are -1, 0, 1 with respect to the eigen vectors (1,-1,0)^{T}, (1,1,-2)^{T}, (1,1,1)^{T}. then 6A is

(a)  \left(\begin{array}{ccc}1 & 5 & 3 \\5 & 1 & 3 \\3 & 3 & 3 \end{array}\right)

(b)  \left(\begin{array}{ccc}-1 & 5 & 2 \\5 & -1 & 2 \\2 & 2 & 2 \end{array}\right)

(c)  \left(\begin{array}{ccc}1 & 1 & 1 \\-1 & 1 & 1 \\0 & -2 & 1 \end{array}\right)

(d)   \left(\begin{array}{ccc}6 & -1 & 0 \\1 & 6 & -2 \\1 & 1 & 6 \end{array}\right)

 

31) The sum of eigen values of \left(\begin{array}{ccc}-1 & -2 & -1 \\-2 & 3 & 2 \\-1 & 2 & -3 \end{array}\right)  is

(a)  -3  (b)  -1  (c)   3  (d)   1

 

32) The matrix \left(\begin{array}{ccc}a^{2} & ab & ac \\ab & b^{2} & bc \\ac & bc & c^{2} \end{array}\right),  where a,b,c\in R-{ 0}  has

(a)  three real, non-zero eigen values  (b)  complex eigen values

(c)  two non-zero eigen values   (d)  only one non-zero eigen value

 

33) \left(\begin{array}{ccc}2 & 0 & 0 \\1 & 2 & 0 \\0 & 0 & 1 \end{array}\right) is

(a)  diagonalizable  (b)  nilpotent  (c)  idempotent   (d)  not diagonalizable

 

34) If a square matrix of order 10 has exactly 5 distinct eigen values, then the degree of the minimal polynomial is

(a)  at least 5  (b)  at most 5  (c)  always 5   (d)   exactly 10

 

35) T: R^{2\times 2}\rightarrow R^{2\times 2}, defined by T(A)=BA, where B=\left(\begin{array}{ccc}1 & -2 \\-2 & 4 \end{array}\right). Then rank of T is

(a)  1  (b)  2  (c)  3   (d)   4

 

36) A=\left(\begin{array}{ccc}1 & a & b \\0 & 10 & c \\0 & 0 & 100 \end{array}\right).  Then

(a)  both A and A^{2} are diagonalizable  (b)  A is diagonalizable but not A^{2}

(c)  A and A^{2} have the same minimal polynomial  (d)  A^{2} is diagonalizable but not A

 

37) Rank of A_{7\times 5} is 5 and that of B_{5\times 7} is 3, then rank of AB is

(a)  1  (b)  2  (c)  3   (d)  4

 

38) A and B are n-square positive definite matrices. Then which of the following are positive definite.

(a)  A+B  (b)  ABA  (c)  AB   (d)   A^{2}+I

 

39) A\in M_{3\times 3}(R) and A=\left(\begin{array}{ccc}2 & 1 & 0 \\0 & 2 & 0 \\0 & 0 & 3 \end{array}\right), then which of the following are subspaces of  M_{3\times 3}(R)

(a) {X\in M_{3\times 3}(R): XA=AX}    (b)  {X\in M_{3\times 3}(R):X+A=A+X}

(c)  {X\in M_{3\times 3}(R): trace(AX)=0}   (d)   {X\in M_{3\times 3}(R): det(AX)=0}

 

40) Let T be a linear operator on the vector space V and T be invariant under the subspace W of V. Then

(a)  T(W)\in W  (b)  W\in T(W)  (c)  T(W)=W   (d)   None of these

 

41) A : R^{4}\rightarrow R^{3}, where  A=\left(\begin{array}{cccc}1 & 2 & 3 & 1 \\1 & 3 & 5 & -2\\3 & 8 & 13 & -3 \end{array}\right). Then the dimension of kernel of A is

READ:   Atheism & Agnosticism

(a)  1  (b)  2  (c)   3  (d)   4

 

42) A : R^{3}\rightarrow R^{4}, where A=\left(\begin{array}{ccc}1 & 1 & 3 \\2 & 3 & 8 \\3 & 5 & 13 \\1 & -2 & -3 \end{array}\right). Then the dimension of image of A is

(a)  1  (b)  2  (c)  3   (d)   4

 

43) Let u, v, w be three non-zero vectors which are linearly independent, then

(a)  u is linear combination of v and w  (b)  v is linear combination of u and w

(c)  w is linear combination of u and v   (d)   none of these

 

44) Let U and W be subspaces of a vector space V and U\cup W is also a subspace of V, then

(a)  either U\subseteq W or W\subseteq U    (b)  U\cap W= \phi   (c)  U=W   (d)   None of these

 

45) Let I be the identity transformation of the finite dimensional vector space V, then the nullity of I is

(a)  dimV  (b)  0  (c)   1  (d)   dimV - 1

 

46) T : R^{3}\rightarrow R^{3}  such that T(a,b,c)=(0,a,b), for (a,b,c)\in R^{3}. Then T+I is a zero of the polynomial:

(a)  t  (b)  t^{2}  (c)   t^{3}  (d)   none of above

 

47) The sum of the eigen values of the matrix  \left(\begin{array}{ccc}4 & 7 & 11 \\7 & 1 & -21\\11 & -21 & 6\end{array}\right) is

(a)  4  (b)  23  (c)  11   (d)   12

 

48) Let A and B are square matrices such that AB=I, then zero is an eigen value of

(a)  A but not of B  (b)  B but not of A  (c)  both A and B   (d)  neither A nor B

 

49) The eigen values of a skew-symmetric matrix are

(a)  negative  (b)  real  (c)  absolute value of 1   (d)   purely imaginary or zero

 

50) The characteristics equation of a matrix A is t^{2}-t-1=0, then

(a)  A^{-1} does not exist (b)  A^{-1} exit but cannot be determined from the data

(c)  A^{-1}=A+1   (d)   A^{-1}=A-1

 

____________________________________________________________________________

The answers can be found here.

 

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Pankaj Jyoti Mahanta

29 Comments
  • Sonia Rode
    Posted at 18:02h, 11 October

    Thanks for the above ques. Can you plz send the correct answers for the same.

    • Gonit Sora
      Posted at 03:14h, 13 October

      Thank you. The answers to these questions will be available shortly.

  • alekhya
    Posted at 22:08h, 26 January

    i need answers to these questions,pls send answers

    • Gonit Sora
      Posted at 01:35h, 07 February

      To view the answers click here.

  • Vidhu Jain
    Posted at 16:29h, 28 March

    Can I get the detailed explaination fo these questions.
    Can u please tell me which book should I use for learning alzebra of matrices or can you give me some notes on the same

    • Gonit Sora
      Posted at 18:48h, 29 March

      A good reference is 'Matrix Analysis'by Horn and Johnson.

    • Gonit Sora
      Posted at 18:50h, 29 March

      And the answers can be found here http://gonitsora.com/answers/

    • SHANMUGAM subramanian
      Posted at 10:29h, 13 August

      Can I get the detailed explaination for these questions.
      Can u please tell me which book should I use for learning alzebra of matrices or can you give me some notes on the same.
      send answer for this gmail: mathsspmanian@gmail.com

  • Gopinath
    Posted at 13:09h, 08 November

    can you plz send these questions above to my email?

    • Gonit Sora
      Posted at 19:32h, 11 November

      The solutions have already been posted elsewhere in this website.

  • satyajit sahoo
    Posted at 17:26h, 11 December

    can you tell me some books for iit jam.

    • Gonit Sora
      Posted at 10:29h, 27 December

      Practise previous years question papers. This is one secret to good JAM score.

      • Saray
        Posted at 23:56h, 27 April

        This is a great way to review and reefrsh. Math, and math homework, were a long time ago but there are still some skills that would be handy to know. Thanks for a great resource.

        • Gonit Sora
          Posted at 12:53h, 12 May

          You are welcome.

  • M. Suganya
    Posted at 11:21h, 19 December

    can u please send these above questions with solutions to my mail

  • Akshayya S. Jadhav
    Posted at 12:34h, 04 January

    Thank You, it's really nice post and also very helpful for me. Thanks again. Please, post another posts also.

    • Gonit Sora
      Posted at 18:41h, 05 January

      We shall post some soon. Thanks for visiting.

  • jaina nithin
    Posted at 20:54h, 13 February

    plz can u post some more questions onlinear algebra
    regarding GATE exam

    • Gonit Sora
      Posted at 12:04h, 21 February

      We are working on it.

  • mayank
    Posted at 14:09h, 23 February

    anwar ple with solution

  • somu.v
    Posted at 05:43h, 20 April

    sir very helpful.Than you sir

  • mabud ali sarkar
    Posted at 19:43h, 03 February

    plz send the questions to my email.id mabudji@gmail.com along with answers

  • Abhinav
    Posted at 23:12h, 09 June

    I think Gonit Sora should post detailed solutions of the questions. All websites provide only answer key. If Gonit Sora publishes full detailed solution , it will be an unique move and immensely helpful to aspirant of competitive exams.

    • Gonit Sora
      Posted at 19:03h, 11 June

      Thank you for the suggestion, we will try to follow it in the future.

  • ravindra
    Posted at 11:02h, 14 June

    good

  • Himanshu tyahi
    Posted at 00:32h, 10 July

    There is no pdf file of some important question ???

    • Gonit Sora
      Posted at 21:29h, 12 July

      Thanks for pointing it out, we have corrected the LaTeX rendering of the formulas.

  • snehal dharmshale
    Posted at 14:01h, 16 September

    Thanks a lot!