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

(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^{ 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^{ 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

 

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

(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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33 Comments
  • Sonia Rode
    Posted at 18:02h, 11 October Reply

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

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

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

  • alekhya
    Posted at 22:08h, 26 January Reply

    i need answers to these questions,pls send answers

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

      To view the answers click here.

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

    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 Reply

      A good reference is ‘Matrix Analysis’by Horn and Johnson.

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

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

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

      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: [email protected]

  • Gopinath
    Posted at 13:09h, 08 November Reply

    can you plz send these questions above to my email?

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

      The solutions have already been posted elsewhere in this website.

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

    can you tell me some books for iit jam.

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

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

      • Saray
        Posted at 23:56h, 27 April Reply

        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 Reply

          You are welcome.

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

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

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

    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 Reply

      We shall post some soon. Thanks for visiting.

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

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

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

      We are working on it.

  • mayank
    Posted at 14:09h, 23 February Reply

    anwar ple with solution

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

    sir very helpful.Than you sir

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

    plz send the questions to my email.id [email protected] along with answers

  • Abhinav
    Posted at 23:12h, 09 June Reply

    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 Reply

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

  • ravindra
    Posted at 11:02h, 14 June Reply

    good

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

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

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

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

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

    Thanks a lot!

  • AJEESH T
    Posted at 20:55h, 31 October Reply

    I think options a and d are correct for 38th question

    • Manjil Saikia
      Posted at 04:10h, 07 November Reply

      Both a and b are correct.

  • Koustav Moni Borah
    Posted at 02:10h, 13 December Reply

    I have doubt about the correct option of question 13 and 29. I think the correct option for question no. 13 is (a) and 29 is (d).

    • Manjil Saikia
      Posted at 11:33h, 28 December Reply

      Hi Koustav, thanks for pointing it out.

      For question 13, can you tell me why this is true?

      For question 29, the answer can be nilpotent because maximum rank for an order n matrix which is nilpotent can be n/2, which is satisfied in this question. I have corrected the answer for this.

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