Linear Algebra GATE Questions

A highly popular method used to prepare for the GATE Exam is to sincerely practise all the previous years’ GATE Questions. Candidates can practise, analyse and understand concepts while solving them. It will also help you strengthen your time management skills. We have attempted to compile, here in this article, a collection of GATE Questions on Linear Algebra.

Candidates are urged to practise these Linear Algebra GATE previous year questions to get the best results. Linear Algebra is an important topic in the GATE CSE question paper, and solving these questions will help the candidates to prepare more proficiently for the GATE exams. Meanwhile, candidates can find the GATE Questions for Linear Algebra here, in this article below, to solve and practise before the exams. They can also refer to these GATE previous year question papers and start preparing for the exams.

GATE Questions on Linear Algebra

1. Let
   \(\begin{array}{l}c_{1}\end{array} \)
   ,………..,
   \(\begin{array}{l}c_{n}\end{array} \)
   be scalars, not all zero, such that
   \(\begin{array}{l}\sum_{i=1}^{n}c_{i}a_{i}=0\end{array} \)
   where
   \(\begin{array}{l}a_{i}\end{array} \)
   are column vectors in
   \(\begin{array}{l}R^{11}\end{array} \)
   . Consider the set of linear equations

Where [

\(\begin{array}{l}a_{1}\end{array} \)
,………..,
\(\begin{array}{l}a_{n}\end{array} \)
] and
\(\begin{array}{l}\sum_{i=1}^{n}a_{i}\end{array} \)

The set of equations has

(GATE CSE 2017 Set 1)

1. A unique solution at
   \(\begin{array}{l}J_{n}\end{array} \)
   where
   \(\begin{array}{l}J_{n}\end{array} \)
   denotes a -dimensional vector of all 1
2. No solution
3. Infinitely many solutions
4. Finitely many solutions

Answer (c)

2. Consider a matrix
   \(\begin{array}{l}A=uv^{T}\end{array} \)
   where
   \(\begin{array}{l}\binom{1}{2}\end{array} \)
   ,
   \(\begin{array}{l}\binom{1}{1}\end{array} \)
   . Note that
   \(\begin{array}{l}v^{T}\end{array} \)
   denotes the

transpose of . The largest eigenvalue of is ________

(GATE CSE 2018)

1. 3
2. 2
3. 1
4. 0

Answer (a)

3. Let X be a square matrix. Consider the following two statements on X.

I. X is invertible.

II. Determinant of X is non-zero.

Which one of the following is TRUE?

(GATE CSE 2019)

1. I implies II; II does not imply I
2. II implies I; I does not imply II
3. I does not imply II; II does not imply I
4. I and II are equivalent statements

Answer (d)

4. The number of divisors of 2100 is ___________

(GATE CSE 2015 Set 2)

1. 36
2. 35
3. 70
4. 30

Answer (a)

5. In the LU decomposition of the matrix
   \(\begin{array}{l}\begin{bmatrix}2 & 2\\4 & 9\\\end{bmatrix}\end{array} \)
   , if the diagonal elements of U are both

1, then the lower diagonal entry

\(\begin{array}{l}l_{22}\end{array} \)
of L is ___________

(GATE CSE 2015 Set 2)

1. 10
2. 5
3. 15
4. 7

Answer (b)

6. Consider the following system of equations:

3x + 2y = 1

4x + 7z = 1

x + y + z =3

x – 2y + 7z = 0

The number of solutions for this system is ____________

(GATE CSE 2014 Set 1)

1. 1
2. 0
3. 0.5
4. None of the above

Answer (a)

7. The value of the dot product of the eigenvectors corresponding to any pair of different eigenvalues of a 4-by-4 symmetric positive definite matrix is ___________

(GATE CSE 2014 Set 1)

1. 1
2. 0
3. 0.5
4. None of the above

Answer (b)

8. Let be the x matrix with elements
   \(\begin{array}{l}a_{11}=a_{12}=a_{21}=+1\end{array} \)
   and
   \(\begin{array}{l}a_{22}=-1\end{array} \)
   . Then the eigenvalues of the matrix are

(GATE CSE 2012)

1. 1024 and 1024
2. 1024 and 1024
3. 4 and 4
4. 512 and 512

Answer (d)

9. The following system of equations
\(\begin{array}{l}x_{1}+x_{2}+2x_{3}=1\end{array} \)
\(\begin{array}{l}x_{1}+2x_{2}+3x_{3}=2\end{array} \)
\(\begin{array}{l}x_{1}+4x_{2}+a2x_{3}=1\end{array} \)

has a unique solution. The only possible value (s) for is/are

(GATE CSE 2008)

1. 0
2. Either 0 or 1
3. One of 0, 1 or -1
4. Any real number except 5

Answer (d)

10. The number of different x symmetric matrices with each elements being either or is

(GATE CSE 2004)

1. \(\begin{array}{l}2^{n}\end{array} \)
2. \(\begin{array}{l}2^{n^{2}}\end{array} \)
3. \(\begin{array}{l}2^{\frac{n^{2}+n}{2}}\end{array} \)
4. \(\begin{array}{l}2^{\frac{n^{2}-n}{2}}\end{array} \)

Answer (c)

11. Consider the following statements:

S1: The sum of two singular n x n matrices may be non-singular

S2: The sum of two n x n non-singular matrices may be singular

Which of the following statements is correct?

(GATE CSE 2001)

1. S1 and S2 are both true
2. S1 is true, S2 is false
3. S1 is false, S2 is true
4. S1 and S2 are both false

Answer (a)

12. An x array v is defined as follows v[i, j] = i – j for all i, j,
    \(\begin{array}{l}1\leq i\leq n\end{array} \)
    ,
    \(\begin{array}{l}1\leq j\leq n\end{array} \)
    . The sum of elements of the array v is

(GATE CSE 2000)

1. 0
2. n
3. \(\begin{array}{l}n^{2}-3n+2\end{array} \)
4. \(\begin{array}{l}n^{2}(n+1)/2\end{array} \)

Answer (a)

13. Consider the following set of equations

x + 2y = 5

4x + 8y = 12

3x + 6y + 3z = 15

This set

(GATE CSE 1998)

1. Has a unique solution
2. Has no solution
3. Has finite number of solutions
4. Has infinite number of solutions

Answer (b)

14. Suppose that the eigenvalues of matrix are.The determinant of
    \(\begin{array}{l}\left ( A^{-1} \right )^{T}\end{array} \)
    is __________

(GATE CSE 2016 Set 2)

1. 0.5
2. 0.125
3. 0.25
4. 0.625

Answer (b)

15. Consider the system, each consisting of m linear equations in variables.

I. If , then all such system have a solution

II. If , then none of these systems has a solution

III. If , then there exists a system which has a solution

Which one of the following is CORRECT?

(GATE CSE 2016 Set 2)

1. I, II, and II are true
2. Only II and III are true
3. Only III is true
4. None of them are true

Answer (c)