﻿ 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

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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 $$T(f(x))=2f^{ ‘}(x)+3 \int_{0}^{x}f(t)dt.$$ 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

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.

Managing Editor of the English Section, Gonit Sora and Research Fellow, Faculty of Mathematics, University of Vienna.

• 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

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

• 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

• 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

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

• 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

• 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.

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