50 questions on linear algebra for NET and GATE aspirants - Gonit Sora (গণিত চ'ৰা)

22 Jun 50 questions on linear algebra for NET and GATE aspirants

Posted at 19:32h in Articles, Careers, English, NET / GATE / SET, Problems by Pankaj Jyoti Mahanta 42 Comments

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

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

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

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

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

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

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

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

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

(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

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

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

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

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

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

(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

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 $A^{2}$

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) ${X\in M_{3\times 3}(R):X+A=A+X}$

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

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

The answers can be found here.

Download this post as PDF (will not include images and mathematical symbols).

Pankaj Jyoti Mahanta

Tags:

characteristics, diagonalizability, eigen value, linear algebra, matrix, rank

42 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.
  - vishnu
    Posted at 17:02h, 24 November Reply
    
    Good job! you done a great work for the new learners in linear algebra like me. But the answer for 29th question is given as option b. But this is not true for the matrix 1 0
    0 0 whose rank is one.
    Here the eigenvalues are 1 and 0 so that this matrix is not nilpotent.
    
    Now, consider the matrix 0 1
    0 0 having rank one. If we note, this matrix is not diagonalizable.
    So, I think the correct option is d.
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
- Manjil Saikia
  Posted at 17:00h, 15 September Reply
  
  Sorry, we do not send emails.
  
  You can get the solutions here: https://gonitsora.com/answers/
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
- Manjil Saikia
  Posted at 17:01h, 15 September Reply
  
  The solutions can be found here: https://gonitsora.com/answers/
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
- Manjil Saikia
  Posted at 17:01h, 15 September Reply
  
  The solutions can be found here: https://gonitsora.com/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
- Gonit Sora
  Posted at 16:01h, 14 June Reply
  
  Thank you.
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.
Akansha tyagi
Posted at 07:56h, 14 September Reply

PDF is including mathematical symbol. Unable to read. How to download it without mathematical symbol
- Manjil Saikia
  Posted at 17:02h, 15 September Reply
  
  At the moment you cannot download the pdf without math symbols. We have corrected the LaTeX rendering of the post now.
Chinnaraj
Posted at 10:46h, 08 August Reply

Can you please send me CSIR UGC NET exam maths questions please sir
Chinnaraj
Posted at 10:47h, 08 August Reply

Can you please send me CSIR UGC NET exam maths questions please sir in my mail
Amber
Posted at 21:00h, 19 February Reply

I think q.29 has only a option correct by matrix first row 10
second 00 then it always come A matrix for any power of A and also it has two distinct e.v 1 and 0 plz consider it so it is diagonalizable not nilpotent