GATE MA topic:linear-algebra PYQs & Answers — Page 5

Home
GATE
Mathematics
Topics
topic:linear-algebra

Questions 41–50 of 62. Imported answers require independent review unless marked reviewed.

Topic practice 62 questionsPage 5 of 7

For a matrix

, the diagonal elements are all zero and

. The minimum number of elements required to fully specify the matrix is _____.

Your answerChoose one option.

You can attempt this question now. Sign in to view the answer key, explanations and discussion.Log in to check answer

Suppose that

is a

matrix with real entries and

. Then

Your answerChoose one option.

every eigenvalue of

is real if

is necessarily a polynomial in

of degree 4 if

is invertible

is not invertible if

You can attempt this question now. Sign in to view the answer key, explanations and discussion.Log in to check answer

Suppose

is a finite dimensional non-zero vector space over

and

is a linear transformation such that

. Then which of the following statements is FALSE?

Your answerChoose one option.

The dimension of

is even

0 is the only eigenvalue of

Both 0 and 1 are eigenvalues of

You can attempt this question now. Sign in to view the answer key, explanations and discussion.Log in to check answer

Let

. Consider the following statements: I: If

for all

and

, then

. II: If

is symmetric and

, then

. Then

Your answerChoose one option.

both I and II are true

I is true but II is false

I is false but II is true

both I and II are false

You can attempt this question now. Sign in to view the answer key, explanations and discussion.Log in to check answer

Suppose

is a finite dimensional vector space over

. If

and

are subspaces of

, then which of the following statements is TRUE?

Your answerChoose one option.

then

and

, then

You can attempt this question now. Sign in to view the answer key, explanations and discussion.Log in to check answer

Suppose that

is a linear transformation over

satisfying

Then the coefficient of

You can attempt this question now. Sign in to view the answer key, explanations and discussion.Log in to check answer

If the characteristic polynomial and minimal polynomial of a square matrix

are

and

, respectively, then the rank of the matrix

is _____, where

is the identity matrix of appropriate order.

You can attempt this question now. Sign in to view the answer key, explanations and discussion.Log in to check answer

Let

be a

real symmetric matrix with eigenvalues

and

with the respective eigenvectors

and

Consider the following statements:

II. The vector

satisfies

III. For any

has a solution.

IV. The trace of the matrix

is 8.

(

denotes the transpose of the vector

)

Which of the above statements are TRUE?

Your answerChoose one option.

I, II and III only

I and II only

II and IV only

III and IV only

You can attempt this question now. Sign in to view the answer key, explanations and discussion.Log in to check answer

Let

be the vector space of all polynomials of degree at most 2 over

(the set of real numbers). Let a linear transformation

be defined by

Consider the following statements.

I. The null space of

II. The range space of

is spanned by the set

III.

IV. If

is the matrix representation of

with respect to the standard basis

, then the trace of the matrix

is 3.

Which of the above statements are TRUE?

Your answerChoose one option.

I and II only

I, III and IV only

I, II and IV only

II and IV only

You can attempt this question now. Sign in to view the answer key, explanations and discussion.Log in to check answer

Let

be the vector space of all

matrices with complex entries over the real field. If

then the dimension of

is equal to ______. (

denotes the conjugate transpose of

)

You can attempt this question now. Sign in to view the answer key, explanations and discussion.Log in to check answer

GATE MA topic:linear-algebra PYQs

Question 9

Question 12

Question 27

Question 28

Question 41

Question 49

Question 20

Question 36

Question 41

Question 46