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

**Find the correct options:**

**1)** and Then is

(a) 0 (b) 1 (c) 2 (d) 3

**2)** where is a 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 is

(a) 1 (b) 2 (c) 3 (d) 4

**4)** The minimal polynomial of is

(a) (b) (c) (d)

**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)** is an operator on The invariant subspaces of the operator are

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

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

**7)** Rank of the matrix is

(a) 2 (b) 3 (c) 4 (d) 5

**8)** The dimension of the subspace of spanned by and is

(a) 1 (b) 2 (c) 3 (d) 4

**9)** U and V are subspace of 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 is

(a) 1 (b) 2 (c) 3 (d) 4

**10)** Let be the set of all n-square symmetric matrices and the characteristics polynomial of each is of the form

Then the dimension of over R is

(a) (b) (c) (d)

**11)** A is a matrix with Then is

(a) 6 (b) 4 (c) 9 (d) 100

**12)** A is 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 is

(a) 1 (b) 2 (c) 3 (d) 4

**15)** A is a matrix over then

(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 Then

(a) B is idempotent (b) exist (c) B is nilpotent (d) B-I is idempotent

**18)** Let then 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 Then

(a) (b) (c) (d)

**20)** and 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 be subset of 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

**22)** such that Then rank of T is

(a) 1 (b) 2 (c) 3 (d) 4

**23)** such that and Then

(a) is 1-1, is not (b) is 1-1, is not

(c) is onto and is 1-1 (d) and both are 1-1

**24)** such that 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)

(b)

(c)

(d) is a basis of

**26)** then

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

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

**27)** defined by 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)** where then

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

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

**29)** 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 then 6A is

(a)

(b)

(c)

(d)

**31)** The sum of eigen values of is

(a) -3 (b) -1 (c) 3 (d) 1

**32)** The matrix where 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)** 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)** defined by T(A)=BA, where Then rank of T is

(a) 1 (b) 2 (c) 3 (d) 4

**36)** Then

(a) both and are diagonalizable (b) is diagonalizable but not

(c) and have the same minimal polynomial (d) is diagonalizable but not

**37)** Rank of is 5 and that of 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)

**39)** and then which of the following are subspaces of

(a) (b)

(c) (d)

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

(a) (b) (c) (d) None of these

**41)** where Then the dimension of kernel of A is

(a) 1 (b) 2 (c) 3 (d) 4

**42)** where 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 is also a subspace of V, then

(a) either or (b) (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)** such that for Then is a zero of the polynomial:

(a) (b) (c) (d) none of above

**47)** The sum of the eigen values of the matrix 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 then

(a) does not exist (b) exit but cannot be determined from the data

(c) (d)

**The answers can be found here.**

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

## Sonia Rode

Posted at 18:02h, 11 OctoberThanks for the above ques. Can you plz send the correct answers for the same.

## Gonit Sora

Posted at 03:14h, 13 OctoberThank you. The answers to these questions will be available shortly.

## alekhya

Posted at 22:08h, 26 Januaryi need answers to these questions,pls send answers

## Gonit Sora

Posted at 01:35h, 07 FebruaryTo view the answers click here.

## vishnu

Posted at 17:02h, 24 NovemberGood 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 MarchCan 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 MarchA good reference is ‘Matrix Analysis’by Horn and Johnson.

## Gonit Sora

Posted at 18:50h, 29 MarchAnd the answers can be found here http://gonitsora.com/answers/

## SHANMUGAM subramanian

Posted at 10:29h, 13 AugustCan 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 Novembercan you plz send these questions above to my email?

## Gonit Sora

Posted at 19:32h, 11 NovemberThe solutions have already been posted elsewhere in this website.

## satyajit sahoo

Posted at 17:26h, 11 Decembercan you tell me some books for iit jam.

## Gonit Sora

Posted at 10:29h, 27 DecemberPractise previous years question papers. This is one secret to good JAM score.

## Saray

Posted at 23:56h, 27 AprilThis 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 MayYou are welcome.

## John

Posted at 21:08h, 10 AprilPlease the site for the solutions

## M. Suganya

Posted at 11:21h, 19 Decembercan u please send these above questions with solutions to my mail

## Manjil Saikia

Posted at 17:00h, 15 SeptemberSorry, we do not send emails.

You can get the solutions here: https://gonitsora.com/answers/

## Akshayya S. Jadhav

Posted at 12:34h, 04 JanuaryThank 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 JanuaryWe shall post some soon. Thanks for visiting.

## jaina nithin

Posted at 20:54h, 13 Februaryplz can u post some more questions onlinear algebra

regarding GATE exam

## Gonit Sora

Posted at 12:04h, 21 FebruaryWe are working on it.

## mayank

Posted at 14:09h, 23 Februaryanwar ple with solution

## Manjil Saikia

Posted at 17:01h, 15 SeptemberThe solutions can be found here: https://gonitsora.com/answers/

## somu.v

Posted at 05:43h, 20 Aprilsir very helpful.Than you sir

## mabud ali sarkar

Posted at 19:43h, 03 Februaryplz send the questions to my email.id [email protected] along with answers

## Manjil Saikia

Posted at 17:01h, 15 SeptemberThe solutions can be found here: https://gonitsora.com/answers/

## Abhinav

Posted at 23:12h, 09 JuneI 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 JuneThank you for the suggestion, we will try to follow it in the future.

## ravindra

Posted at 11:02h, 14 Junegood

## Gonit Sora

Posted at 16:01h, 14 JuneThank you.

## Himanshu tyahi

Posted at 00:32h, 10 JulyThere is no pdf file of some important question ???

## Gonit Sora

Posted at 21:29h, 12 JulyThanks for pointing it out, we have corrected the LaTeX rendering of the formulas.

## snehal dharmshale

Posted at 14:01h, 16 SeptemberThanks a lot!

## AJEESH T

Posted at 20:55h, 31 OctoberI think options a and d are correct for 38th question

## Manjil Saikia

Posted at 04:10h, 07 NovemberBoth a and b are correct.

## Koustav Moni Borah

Posted at 02:10h, 13 DecemberI 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 DecemberHi 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 SeptemberPDF is including mathematical symbol. Unable to read. How to download it without mathematical symbol

## Manjil Saikia

Posted at 17:02h, 15 SeptemberAt 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 AugustCan you please send me CSIR UGC NET exam maths questions please sir

## Chinnaraj

Posted at 10:47h, 08 AugustCan you please send me CSIR UGC NET exam maths questions please sir in my mail

## Amber

Posted at 21:00h, 19 FebruaryI 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

## Ritu

Posted at 09:34h, 20 MayI think for question 21 option d is also correct