
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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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.
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