Linear Algebra - Engineering Mathematics

Q#1 Linear Algebra GATE EE 2026 (Set 1) MSQ +1 mark -0 marks

Two matrices and have a common eigenvalue 2, and the same corresponding nonzero eigenvector.

Which of the following options is/are correct?

(Note: is the identity matrix.)

Determinant

Q#2 Linear Algebra GATE EE 2026 (Set 1) NAT +1 mark -0 marks

is an skew-symmetric matrix with real-valued entries, and is an -dimensional column vector with real-valued entries such that .

The quantity evaluates to

(Answer in integer)

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Q#3 Linear Algebra GATE EE 2026 (Set 1) MCQ +2 marks -0.66 marks

Which one of the following statements is ALWAYS correct about a collection of column vectors, each having real-valued entries?

If , then the column vectors must be linearly dependent

If , then the column vectors must be linearly independent

If , then the column vectors must be orthogonal

If , then the column vectors must be linearly independent

Q#4 Linear Algebra GATE EE 2026 (Set 1) MSQ +2 marks -0 marks

Consider an orthogonal matrix with real entries and each column having unit Euclidean norm.

Which of the following statements is/are correct?

The value of the determinant of is either +1 or -1

The eigenvalues of have modulus 1

, for all , where denotes the Euclidean norm of , and , for all distinct

, for all , where denotes the Euclidean norm of , and , for all

Q#5 Linear Algebra GATE EE 2026 (Set 1) MSQ +2 marks -0 marks

Consider the system of linear equations: , where is an matrix, and and are -dimensional column vectors.

Suppose this system of equations has a unique solution. Which of the following statements is/are correct?

exists

The system of equations also has a unique solution for

, for

, where denotes the augmented matrix.

Q#6 Linear Algebra GATE EE 2025 (Set 1) MCQ +1 mark -0.33 marks

Consider the set of points which minimize the real valued function

Which of the following statements is true about the set S?

The number of elements in the set S is finite and more than one.

The number of elements in the set S is infinite.

The set S is empty.

The number of elements in the set S is exactly one.

Q#7 Linear Algebra GATE EE 2025 (Set 1) MCQ +1 mark -0.33 marks

Let and be the two eigenvectors corresponding to distinct eigenvalues of a real symmetric matrix. Which one of the following statements is true?

Q#8 Linear Algebra GATE EE 2025 (Set 1) MCQ +1 mark -0.33 marks

Let , and . Then, the system of linear equations has

a unique solution.

infinitely many solutions.

a finite number of solutions.

no solution.

Q#9 Linear Algebra GATE EE 2025 (Set 1) MCQ +1 mark -0.33 marks

Let and let I can be identity matrix. Then is equal to

2P-I

P

I

P+I

Q#10 Linear Algebra GATE EE 2024 (Set 1) MCQ +1 mark -0.33 marks

Which one of the following matrices has an inverse?

Q#11 Linear Algebra GATE EE 2024 (Set 1) NAT +1 mark -0 marks

The sum of the eigenvalues of the matrix is ________ (rounded off to the nearest integer).

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Q#12 Linear Algebra GATE EE 2023 (Set 1) MCQ +1 mark -0.33 marks

For a given vector , the vector normal to the plane defined by is

Q#13 Linear Algebra GATE EE 2023 (Set 1) MCQ +1 mark -0.33 marks

In the figure, the vectors and are related as: by a transformation matrix A. The correct choice of is

Q#14 Linear Algebra GATE EE 2022 (Set 1) MCQ +1 mark -0.33 marks

Consider a matrix whose -th element, . Then the matrix will be

symmetric

skew-symmetric

unitary

null

Q#15 Linear Algebra GATE EE 2022 (Set 1) MCQ +2 marks -0.66 marks

denotes the exponential of a square matrix . Suppose is an eigen value and is the corresponding eigen-vector of matrix .

Consider the following two statements:

Statement 1: is an eigen value of .

Statement 2: is an eigen-vector of .

Which one of the following options is correct?

Statement 1 is true and statement 2 is false.

Statement 1 is false and statement 2 is true.

Both the statements are correct.

Both the statements are false.

Q#16 Linear Algebra GATE EE 2022 (Set 1) MCQ +2 marks -0.66 marks

Consider a matrix . The matrix satisfies the equation , where and are scalars and is the identity matrix. Then is equal to

5

17

-6

11

Q#17 Linear Algebra GATE EE 2021 (Set 1) MCQ +1 mark -0.33 marks

Let and be real numbers such that .

The eigenvalues of the matrix are

pq and –pq

1 and 1

and

1 and -1

Q#18 Linear Algebra GATE EE 2021 (Set 1) NAT +2 marks -0 marks

Let be a matrix such that is null matrix, and let be the identity matrix. The determinant of is _______.

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Q#19 Linear Algebra GATE EE 2020 (Set 1) MSQ +2 marks -0.66 marks

The number of purely real elements in a lower triangular representation of the given matrix, obtained through the given decomposition is

6

5

8

9

Q#20 Linear Algebra GATE EE 2019 (Set 1) NAT +1 mark -0 marks

The rank of the matrix, is __________________.

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Q#21 Linear Algebra GATE EE 2019 (Set 1) MCQ +1 mark -0.33 marks

M is a 2 × 2 matrix with eigenvalues 4 and 9. The eigenvalues of are

2 and 3

16 and 81

–2 and –3

4 and 9

Q#22 Linear Algebra GATE EE 2019 (Set 1) MCQ +2 marks -0.66 marks

Consider a matrix , where, and are the column vectors. Suppose , where and are the row vectors. Consider the following statements:

Statement 1:

Statement 2:

Statement 1 is true and statement 2 is false

Statement 2 is true and statement 1 s false

Both the statements are true

Both the statements are false

Q#23 Linear Algebra GATE EE 2018 (Set 1) NAT +1 mark -0 marks

Consider a non-singular square matrix A If and, the determinant of the matrix A is ________ (up to 1 decimal place).

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Q#24 Linear Algebra GATE EE 2018 (Set 1) NAT +2 marks -0 marks

Let and, where I is identity matrix. The determinant of B is ______ (up to 1 decimal place).

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Q#25 Linear Algebra GATE EE 2017 (Set 2) MCQ +2 marks -0.66 marks

The eigen values of the matrix given below are

.

(0,-1,-3)

(0,-2,-3)

(0,2,3)

(0,1,3)

Q#26 Linear Algebra GATE EE 2017 (Set 1) MCQ +1 mark -0.33 marks

The matrix has three distinct eigenvalues and one of its eigenvectors is . Which one of the following can be another eigenvector of A?

Q#27 Linear Algebra GATE EE 2017 (Set 2) NAT +1 mark -0 marks

Let and . The value of equals _________. (Give the answer up to three decimal places)

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Q#28 Linear Algebra GATE EE 2016 (Set 1) NAT +1 mark -0 marks

Consider a matrix with every element being equal to 1. Its only non-zero Eigen value is ___________.

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Q#29 Linear Algebra GATE EE 2016 (Set 1) MCQ +2 marks -0.66 marks

Let the Eigen values of a matrix A be 1, -2 with eigenvectors and respectively. Then the Eigen values and eigenvectors of the matrix would, respectively, be

Q#30 Linear Algebra GATE EE 2016 (Set 1) MCQ +2 marks -0.66 marks

Let A be a real matrix with rank 2. Which one of the following statement is TRUE?

Rank of is less than 2

Rank of is equal to 2

Rank of is greater than 2

Rank of can be any number between 1 and 3

Q#31 Linear Algebra GATE EE 2016 (Set 2) MCQ +1 mark -0.33 marks

A matrix P is such that, . Then the eigenvalues of P are

1, 1, -1

1, 0.5, +j0.866, 0.5 – j0.866

1, -0.5 + j0.866, -0.5 – j0.866

0, 1, -1

Q#32 Linear Algebra GATE EE 2016 (Set 2) MCQ +2 marks -0.66 marks

Let . Consider the set S of all vectors such that where . Then S is

a circle of radius

an ellipse with major axis along

an ellipse with minor axis along

Q#33 Linear Algebra GATE EE 2015 (Set 1) NAT +1 mark -0 marks

If the sum of the diagonal elements of a 2 x 2 matrix is -6, then the maximum possible value of determinant of the matrix is ____________.

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Q#34 Linear Algebra GATE EE 2015 (Set 1) MCQ +2 marks -0.66 marks

The maximum value of ‘a’ such that the matrix has three linearly independent real eigenvectors is

Q#35 Linear Algebra GATE EE 2015 (Set 2) MCQ +1 mark -0.33 marks

We have a set of 3 linear equations in 3 unknowns. ‘X = Y’ means X and Y are equivalent statements and means X and Y are not equivalent statements.

P. There is a unique solution.

Q. The equations are linearly independent.

R. All Eigen values of the coefficient matrix are nonzero.

S. The determinant of the coefficient matrix is nonzero.

Which one of the following is TRUE?

Q#36 Linear Algebra GATE EE 2014 (Set 1) MCQ +1 mark -0.33 marks

Given a system of equations:

Which of the following is true regarding its solutions

The system has a unique solution for any given

The system will have infinitely many solutions for any given

Whether or not a solution exists depends on the given

The system would have no solution for any values of

Q#37 Linear Algebra GATE EE 2014 (Set 1) NAT +2 marks -0 marks

A system matrix is given as follows.

The absolute value of the ratio of the maximum Eigen value to the minimum Eigen value is ______________.

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Q#38 Linear Algebra GATE EE 2014 (Set 2) MCQ +1 mark -0.33 marks

Which one of the following statements is true for all real symmetric matrices?

All the Eigen values are real.

All the Eigen values are positive.

All the Eigen values are distinct.

Sum of all the Eigen values is zero.

Q#39 Linear Algebra GATE EE 2014 (Set 3) MCQ +1 mark -0.33 marks

Two matrices A and B are given below:

If the rank of matrix A is N, then the rank of matrix B is

N/2

N-1

N

2 N

Q#40 Linear Algebra GATE EE 2013 (Set 1) MCQ +1 mark -0.33 marks

The equation has

No solution

Only one solution

Non-zero unique solution

Multiple solutions

Q#41 Linear Algebra GATE EE 2013 (Set 1) MCQ +2 marks -0.66 marks

A matrix has Eigen values –1 and -2. The corresponding eigenvectors are and respectively. The matrix is

Q#42 Linear Algebra GATE EE 2012 (Set 1) MCQ +2 marks -0.66 marks

Given that

and , the value of is

15A + 12I

19A + 30I

17A + 15I

17A + 21I

Q#43 Linear Algebra GATE EE 2011 (Set 1) MCQ +2 marks -0.66 marks

The matrix is decomposed into a product of a lower triangular matrix [L] and an upper triangular matrix [U]. The properly decomposed [L] and [U] matrices respectively are

Q#44 Linear Algebra GATE EE 2010 (Set 1) MCQ +2 marks -0.66 marks

An eigenvector of is

Q#45 Linear Algebra GATE EE 2010 (Set 1) MCQ +2 marks -0.66 marks

For the set of equations

The following statement is true:

Only the trivial solution exists.

There are no solutions.

A unique non-trivial solution exists

Multiple non-trivial solutions exist

Q#46 Linear Algebra GATE EE 2009 (Set 1) MCQ +1 mark -0.33 marks

The trace and determinant of a 2 × 2 matrix are known to be -2 and -35 respectively. Its Eigen values are

−30 and 5

−37 and −1

−7 and 5

17.5 and −2

Q#47 Linear Algebra GATE EE 2008 (Set 1) MCQ +1 mark -0.33 marks

The characteristic equation of a (3 × 3) matrix P is defined as. If I denote identity matrix, then the inverse of matrix P will be

Q#48 Linear Algebra GATE EE 2008 (Set 1) MCQ +1 mark -0.33 marks

If the rank of a (5×6) matrix Q is 4, then which one of the following statements is correct?

Q will have four linearly independent rows and four linearly Independent columns

Q will have four linearly Independent rows and five linearly Independent columns

will be invertible

Q#49 Linear Algebra GATE EE 2008 (Set 1) MCQ +2 marks -0.66 marks

A is a m×n full ran matrix with m > n and I is an identity matrix. Let matrix. Then, which one of the following statements is FALSE?

Q#50 Linear Algebra GATE EE 2008 (Set 1) MCQ +2 marks -0.66 marks

Let P be a 2 ´ 2 real orthogonal matrix and is a real vector with length. Then, which one of the following statements is correct?

where at least one vector satisfies

for all vectors

where at least one vector satisfies

No relationship can be established between and

Q#51 Linear Algebra GATE EE 2007 (Set 1) MCQ +1 mark -0.33 marks

is an n-tuple nonzero vector. The n × n matrix

Has rank zero

Has rank 1

Is orthogonal

Has rank n

Q#52 Linear Algebra GATE EE 2007 (Set 1) MCQ +2 marks -0.66 marks

Let x and y be two vectors in a 3 dimensional space and <x, y> denote their dot product. Then the determinant

Is zero when x and y are linearly independent

Is positive when x and y are linearly independent

Is non-zero for all non-zero x and y

Is zero only when either x or y is zero

Q#53 Linear Algebra GATE EE 2007 (Set 1) MCQ +2 marks -0.66 marks

The linear operation L(x) is defined by the cross product , where and are three dimensional vectors. The 3×3 matrix M of this operation satisfies

Then the Eigen values of M are

0, +1, -1

1, -1, 1

i, −i, 1

i, −i, 0

Q#54 Linear Algebra GATE EE 2007 (Set 1) MSQ +2 marks -0 marks

Cayley-Hamilton Theorem states that a square matrix satisfies its own characteristic equation.

Consider a matrix

A satisfies the relation

Q#55 Linear Algebra GATE EE 2007 (Set 1) MCQ +2 marks -0.66 marks

Cayley-Hamilton Theorem states that a square matrix satisfies its own characteristic equation.

Consider a matrix

equals

Q#56 Linear Algebra GATE EE 2006 (Set 1) MCQ +2 marks -0.66 marks

are three vectors

An orthogonal set of vectors having a span that contains p, q, r is

Q#57 Linear Algebra GATE EE 2006 (Set 1) MCQ +2 marks -0.66 marks

are three vectors

The following vector is linearly dependent upon the solution to the previous problem

Q#58 Linear Algebra GATE EE 2005 (Set 1) MCQ +1 mark -0.33 marks

In the matrix equation Px = q, which of the following is a necessary condition for the existence of at least one solution for the unknown vector x:

Augmented matrix [Pq] must have the same rank as matrix P

Vector q must have only non-zero elements

Matrix P must be singular

Matrix P must be square

Q#59 Linear Algebra GATE EE 2005 (Set 1) MCQ +2 marks -0.66 marks

For the matrix , one of the Eigen values is equal to –2. Which of the following is an Eigen vector?

Q#60 Linear Algebra GATE EE 2005 (Set 1) MCQ +2 marks -0.66 marks

If , the top row of is:

Q#61 Linear Algebra GATE EE 2002 (Set 1) MCQ +1 mark -0.33 marks

The determinant of the matrix

is:

100

200

1

300

Q#62 Linear Algebra GATE EE 1998 (Set 1) MCQ +1 mark -0.33 marks

A set of linear equations is represented by the matrix equation Ax=b the necessary condition for the existence of a solution for this system is:

A must be invertible

B must be linearly depend on the columns of A

B must be linearly independent of the columns of A

None of the above

Q#63 Linear Algebra GATE EE 1998 (Set 1) MCQ +1 mark -0.33 marks

The vector is an Eigen vector of . One of the given values of A is:

1

2

5

–1

Q#64 Linear Algebra GATE EE 1998 (Set 1) MCQ +1 mark -0.33 marks

The sum of the Eigen values of the matrix A is:

10

–10

24

22

Q#65 Linear Algebra GATE EE 1998 (Set 1) MCQ +2 marks -0.66 marks

The inverse of A is:

Q#66 Linear Algebra GATE EE 1997 (Set 1) MCQ +1 mark -0.33 marks

A square matrix is called singular, if its

Determinant is unity

Determinant is zero

Determinant is infinity

Rank is unity

Q#67 Linear Algebra GATE EE 1997 (Set 1) MCQ +1 mark -0.33 marks

Gauss-Seidel iterative method can be used for solving a set of

Linear differential equations only

Linear algebraic equations only

Both linear and nonlinear algebraic equation

Both linear and non linear differential equations

Q#68 Linear Algebra GATE EE 1997 (Set 1) MCQ +2 marks -0.66 marks

Express the given matrix A as a product of two triangular matrices, L and U, where the diagonal elements of the lower triangular matrix L are unity and U is an upper triangular matriX