## 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.**

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#### Pankaj Jyoti Mahanta

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## 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.

## 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: mathsspmanian@gmail.com

## 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.

## M. Suganya

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

## 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

## 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 mabudji@gmail.com along with 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.