GATE MA topic:linear-algebra PYQs & Answers

Questions 1–10 of 62. Imported answers require independent review unless marked reviewed.

Let

be the vector space of all

real matrices over

under usual matrix addition and scalar multiplication. Let

be the set of all

real upper triangular matrices. Which one of the following is TRUE?

Your answerChoose one option.

The quotient space

is isomorphic to the vector space of all

real skew symmetric matrices over

under usual matrix addition and scalar multiplication.

The quotient space

is isomorphic to the vector space of all

real symmetric matrices over

under usual matrix addition and scalar multiplication.

The quotient space

is isomorphic to the vector space of all

real lower triangular matrices over

under usual matrix addition and scalar multiplication.

The quotient space

is isomorphic to the vector space of all

real matrices with trace zero over

under usual matrix addition and scalar multiplication.

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Let

be a

real symmetric positive definite matrix. Which of the following statements is/are TRUE?

Your answerSelect all correct options. You can choose more than one.

is positive definite for all nonzero

is positive semidefinite for all

is positive definite if

is nonsingular.

is not positive semidefinite for some

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Let

be the linear transformation which reflects every vector in

through a two-dimensional subspace of

. Let

be the matrix representation of

using the basis

. Then the value of

is equal to ________. (answer in integer)

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Let

. Define an inner product on

with respect to

. Consider the subspace

. Which one of the following sets is an orthonormal basis of

with respect to

Your answerChoose one option.

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Let

and

nonzero real matrices. Assume that there exists a

real nonsingular matrix

such that

and

are both upper triangular. Which of the following statements is/are TRUE?

Your answerSelect all correct options. You can choose more than one.

The matrix

is invertible.

The matrix

is nilpotent.

The matrix

is invertible, where

is the

identity matrix.

is a nonzero matrix, then

is diagonalizable.

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Let

. Then the geometric multiplicity of the eigenvalue 2 of

is equal to ________. (answer in integer)

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Consider the problem of maximizing

subject to

Then the maximum value of

is ________. (answer in integer)

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Let

be the vector space of all polynomials of degree at most three with the real coefficients under usual polynomial addition and scalar multiplication. Let

be the linear transformation defined as

for all

, where

is the derivative of

. Then the nullity of

is equal to ________. (answer in integer)

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Consider the system of ordinary differential equations

where

is a

skew-symmetric matrix with entries in

. Then, for this system, the origin is a stable critical point for

Your answerChoose one option.

any such matrix

only such matrices

whose rank is 2

only such matrices

whose rank is 4

only such matrices

whose rank is 6

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Let

denote the characteristic polynomial of a square matrix

. Then, for which of the following invertible matrices

, the polynomial

is constant?

Your answerSelect all correct options. You can choose more than one.

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GATE MA topic:linear-algebra PYQs

How to practice topic:linear-algebra

Question 11

Question 21

Question 28

Question 37

Question 48

Question 55

Question 57

Question 58

Question 16

Question 23