Linear Algebra

Questions

1. GATE CS 2026 Set 1MCQ1 markEasy (estimate)

   GATE CS 2026 Set 1, Q.13

   For $n>1$, the maximum multiplicity of any eigenvalue of an $n\times n$ matrix with elements from $\mathbb{R}$ is

   Linear Algebra

2. GATE CS 2026 Set 1MSQ1 markEasy (estimate)

   GATE CS 2026 Set 1, Q.20

   Let $n>1$. Consider an $n\times n$ matrix $M$ with its elements from $\mathbb{R}$. Let the vector $(0,1,0,0,\dots ,0)\in {\mathbb{R}}^n$ be in the null space of $M$. Which...

   Linear Algebra

3. GATE CS 2026 Set 2NAT1 markEasy (estimate)

   GATE CS 2026 Set 2, Q.31

   Consider the system of linear equations given below. $ax+y=b$ $16x+ay=24$ Suppose the values of $a$ and $b$ are chosen such that the system of linear equations produ...

   Linear Algebra

4. GATE CS 2026 Set 2NAT2 marksEasy (estimate)

   GATE CS 2026 Set 2, Q.62

   The determinant of a $4\times 4$ matrix $A$ is $3$. The value of the determinant of $2A$ is ____________. (answer in integer)

   Linear Algebra

5. GATE CS 2025 Set 1MSQ1 markEasy (estimate)

   GATE CS 2025 Set 1, Q.23

   Consider the given system of linear equations for variables $x$ and $y$, where $k$ is a real-valued constant. Which of the following option(s) is/are CORRECT? $x+ky=1$…

   Linear Algebra

6. GATE CS 2025 Set 1MCQ2 marksEasy (estimate)

   GATE CS 2025 Set 1, Q.41

   Let $A$ be a $2\times 2$ matrix as given. $A=\left[\begin{array}{cc}1& 1\\ 1& -1\end{array}\right]$ What are the eigenvalues of the matrix $A^{13}$?

   Linear Algebra

7. GATE CS 2025 Set 2MCQ1 markEasy (estimate)

   GATE CS 2025 Set 2, Q.11

   If $A=\left(\begin{array}{cc}1& 2\\ 2& -1\end{array}\right)$, then which ONE of the following is $A^8$?

   Linear Algebra

8. GATE CS 2025 Set 2MCQ1 markEasy (estimate)

   GATE CS 2025 Set 2, Q.14

   Let $L,M,$ and $N$ be non-singular matrices of order 3 satisfying the equations $L^2=L^{-1},M=L^8$ and $N=L^2$. Which ONE of the following is the value of the determi...

   Linear Algebra

9. GATE CS 2025 Set 2MSQ2 marksEasy (estimate)

   GATE CS 2025 Set 2, Q.44

   Consider a system of linear equations $PX=Q$ where $P\in {\mathbb{R}}^{3\times 3}$ and $Q\in {\mathbb{R}}^{3\times 1}$. Suppose $P$ has an LU decomposition, $P=LU$, where…

   Linear Algebra

10. GATE CS 2024 Set 1MCQ1 markEasy (estimate)

    GATE CS 2024 Set 1, Q.12

    The product of all eigenvalues of the matrix $\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right]$ is

    Linear Algebra

11. GATE CS 2024 Set 1MSQ2 marksHard (estimate)

    GATE CS 2024 Set 1, Q.49

    Let $A$ be any $n\times m$ matrix, where $m>n$. Which of the following statements is/are TRUE about the system of linear equations $Ax=0$ ?

    Linear Algebra

12. GATE CS 2024 Set 2MCQ1 markMedium (estimate)

    GATE CS 2024 Set 2, Q.17

    Let $A$ be the adjacency matrix of a simple undirected graph $G$. Suppose $A$ is its own inverse. Which one of the following statements is always TRUE?

    Graph Theory · Linear Algebra

13. GATE CS 2024 Set 2MSQ2 marksEasy (estimate)

    GATE CS 2024 Set 2, Q.47

    Let $A$ be an $n\times n$ matrix over the set of all real numbers $\mathbb{R}$. Let $B$ be a matrix obtained from $A$ by swapping two rows. Which of the following statements is...

    Linear Algebra

14. GATE CS 2023MCQ1 markEasy (estimate)

    GATE CS 2023, Q.18

    Let $A=\left[\begin{array}{cccc}1& 2& 3& 4\\ 4& 1& 2& 3\\ 3& 4& 1& 2\\ 2& 3& 4& 1\end{array}\right]$ and…

    Linear Algebra

15. GATE CS 2023NAT1 markMedium (estimate)

    GATE CS 2023, Q.30

    Let $A$ be the adjacency matrix of the graph with vertices $\{1,2,3,4,5\}$. Undirected graph with 5 vertices {1, 2, 3, 4, 5}, edges (1,2), (2,3), (2,4), (3,5), (4,5), and se...

    Linear Algebra · Graph Theory

16. GATE CS 2022MCQ1 markEasy (estimate)

    GATE CS 2022, Q.20

    Consider the following two statements with respect to the matrices $A_{m\times n}$, $B_{n\times m}$, $C_{n\times n}$ and $D_{n\times n}$. Statement 1…

    Linear Algebra

17. GATE CS 2022MCQ2 marksMedium (estimate)

    GATE CS 2022, Q.45

    Consider solving the following system of simultaneous equations using LU decomposition. $x_1+x_2-2x_3=4$ $x_1+3x_2-x_3=7$ $2x_1+x_2-5x_3=7$ where L and U...

    Linear Algebra

18. GATE CS 2022MSQ2 marksMedium (estimate)

    GATE CS 2022, Q.53

    Which of the following is/are the eigenvector(s) for the matrix given below?…

    Linear Algebra

19. GATE CS 2021 Set 1NAT2 marksEasy (estimate)

    GATE CS 2021 Set 1, Q.62 (CS Q.52)

    Consider the following matrix. $\left(\begin{array}{cccc}0& 1& 1& 1\\ 1& 0& 1& 1\\ 1& 1& 0& 1\\ 1& 1& 1& 0\end{array}\right)$ The largest eigenvalue of the above matrix is _...

    Linear Algebra

20. GATE CS 2021 Set 2NAT1 markEasy (estimate)

    GATE CS 2021 Set 2, Q.34 (CS Q.24)

    Suppose that $P$ is a $4\times 5$ matrix such that every solution of the equation $Px=0$ is a scalar multiple of $[25431{]}^T$. The rank of $P$ is ______.

    Linear Algebra

21. GATE CS 2021 Set 2MCQ2 marksEasy (estimate)

    GATE CS 2021 Set 2, Q.47 (CS Q.37)

    For two $n$-dimensional real vectors $P$ and $Q$, the operation $s(P,Q)$ is defined as follows: $s(P,Q)={\sum }_{i=1}^n(P_i\cdot Q_i)$ Let $\mathcal{L}$ be a set of 10-dime...

    Linear Algebra

22. GATE CS 2020MCQ2 marksEasy (estimate)

    GATE CS 2020, Q.37 (CS Q.27)

    Let $A$ and $B$ be two $n\times n$ matrices over real numbers. Let $\text{rank}(M)$ and $\det (M)$ denote the rank and determinant of a matrix $M$, respectively. Consider the fo...

    Linear Algebra

23. GATE CS 2019MCQ1 markEasy (estimate)

    GATE CS 2019, Q.19 (CS Q.9)

    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?

    Linear Algebra

24. GATE CS 2019NAT2 marksMedium (estimate)

    GATE CS 2019, Q.54 (CS Q.44)

    Consider the following matrix: $R=\left[\begin{array}{cccc}1& 2& 4& 8\\ 1& 3& 9& 27\\ 1& 4& 16& 64\\ 1& 5& 25& 125\end{array}\right]$ The absolute value of the product of...

    Linear Algebra

25. GATE CS 2018NAT1 markEasy (estimate)

    GATE CS 2018, Q.27 (CS Q.17)

    Consider a matrix $A=u{v}^T$ where $u=\left(\begin{array}{c}1\\ 2\end{array}\right),v=\left(\begin{array}{c}1\\ 1\end{array}\right)$. Note that $v^T$ denotes the transpose of $v$. The large...

    Linear Algebra

26. GATE CS 2018MCQ2 marksMedium (estimate)

    GATE CS 2018, Q.36 (CS Q.26)

    Consider a matrix $P$ whose only eigenvectors are the multiples of $\left[\begin{array}{c}1\\ 4\end{array}\right]$. Consider the following statements: (I) $P$ does not have an inverse (I...

    Linear Algebra

27. GATE CS 2017 Set 1MCQ1 markEasy (estimate)

    GATE CS 2017 Set 1, Q.3 (CS Q.3)

    Let $c_1,\dots ,c_n$ be scalars, not all zero, such that ${\sum }_{i=1}^n{c}_i{a}_i=0$ where $a_i$ are column vectors in ${\mathbb{R}}^n$. Consider the set of linear equations…

    Linear Algebra

28. GATE CS 2017 Set 1MCQ2 marksEasy (estimate)

    GATE CS 2017 Set 1, Q.30 (CS Q.30)

    Let $u$ and $v$ be two vectors in ${\mathbb{R}}^2$ whose Euclidean norms satisfy $\Vert u\Vert =2\Vert v\Vert$. What is the value of $\alpha$ such that $w=u+\alpha v$ bisects the angle be...

    Linear Algebra

29. GATE CS 2017 Set 1MCQ2 marksMedium (estimate)

    GATE CS 2017 Set 1, Q.31 (CS Q.31)

    Let $A$ be an $n\times n$ real valued square symmetric matrix of rank 2 with ${\sum }_{i=1}^n{\sum }_{j=1}^n{A}_{ij}^2=50$. Consider the following statements: (I) One eigenvalue mu...

    Linear Algebra

30. GATE CS 2017 Set 2NAT1 markEasy (estimate)

    GATE CS 2017 Set 2, Q.22 (CS Q.22)

    Let $P=\left[\begin{array}{ccc}1& 1& -1\\ 2& -3& 4\\ 3& -2& 3\end{array}\right]$ and $Q=\left[\begin{array}{ccc}-1& -2& -1\\ 6& 12& 6\\ 5& 10& 5\end{array}\right]$ be two matrices....

    Linear Algebra

31. GATE CS 2017 Set 2NAT2 marksEasy (estimate)

    GATE CS 2017 Set 2, Q.52 (CS Q.52)

    If the characteristic polynomial of a $3\times 3$ matrix $M$ over $\mathbb{R}$ (the set of real numbers) is ${\lambda }^3-4{\lambda }^2+a\lambda +30$, $a\in \mathbb{R}$, and on...

    Linear Algebra

32. GATE CS 2016 Set 1NAT1 markEasy (estimate)

    GATE CS 2016 Set 1, Q.15 (CS Q.5)

    Two eigenvalues of a $3\times 3$ real matrix $P$ are $(2+\sqrt{-1})$ and $3$. The determinant of $P$ is _____.

    Linear Algebra

33. GATE CS 2016 Set 2MCQ1 markEasy (estimate)

    GATE CS 2016 Set 2, Q.14 (CS Q.4)

    Consider the systems, each consisting of $m$ linear equations in $n$ variables. I. If $m<n$, then all such systems have a solution II. If $m>n$, then none of these systems h...

    Linear Algebra

34. GATE CS 2016 Set 2NAT1 markEasy (estimate)

    GATE CS 2016 Set 2, Q.16 (CS Q.6)

    Suppose that the eigenvalues of matrix $A$ are 1, 2, 4. The determinant of $(A^{-1}{)}^T$ is ________.

    Linear Algebra

35. GATE CS 2015 Set 1NAT1 markEasy (estimate)

    GATE CS 2015 Set 1, Q.28 (CS Q.18)

    In the LU decomposition of the matrix $\left[\begin{array}{cc}2& 2\\ 4& 9\end{array}\right]$, if the diagonal elements of $U$ are both $1$, then the lower diagonal entry $l_{22}$ of $L$...

    Linear Algebra

36. GATE CS 2015 Set 1MCQ2 marksEasy (estimate)

    GATE CS 2015 Set 1, Q.46 (CS Q.36)

    Consider the following $2\times 2$ matrix $A$ where two elements are unknown and are marked by $a$ and $b$. The eigenvalues of this matrix are $-1$ and $7$. What are the values...

    Linear Algebra

37. GATE CS 2015 Set 2NAT1 markEasy (estimate)

    GATE CS 2015 Set 2, Q.15 (CS Q.5)

    The larger of the two eigenvalues of the matrix $\left[\begin{array}{cc}4& 5\\ 2& 1\end{array}\right]$ is _____.

    Linear Algebra

38. GATE CS 2015 Set 2NAT2 marksEasy (estimate)

    GATE CS 2015 Set 2, Q.37 (CS Q.27)

    Perform the following operations on the matrix $\left[\begin{array}{ccc}3& 4& 45\\ 7& 9& 105\\ 13& 2& 195\end{array}\right]$: (i) Add the third row to the second row. (ii) Subtract...

    Linear Algebra

39. GATE CS 2015 Set 3MCQ1 markEasy (estimate)

    GATE CS 2015 Set 3, Q.25 (CS Q.15)

    In the given matrix $\left[\begin{array}{ccc}1& -1& 2\\ 0& 1& 0\\ 1& 2& 1\end{array}\right]$ one of the eigenvalues is 1. The eigenvectors corresponding to the eigenvalue 1 are

    Linear Algebra

40. GATE CS 2015 Set 3MCQ2 marksMedium (estimate)

    GATE CS 2015 Set 3, Q.43 (CS Q.33)

    If the following system has non-trivial solution, $\begin{array}{rl}px+qy+rz& =0\\ qx+ry+pz& =0\\ rx+py+qz& =0\end{array}$ then which one of the...

    Linear Algebra

41. GATE CS 2014 Set 1NAT1 markMedium (estimate)

    GATE CS 2014 Set 1, Q.14 (CS Q.4)

    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 __________________

    Linear Algebra

42. GATE CS 2014 Set 1NAT1 markEasy (estimate)

    GATE CS 2014 Set 1, Q.15 (CS Q.5)

    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 _____________________.

    Linear Algebra

43. GATE CS 2014 Set 2NAT1 markEasy (estimate)

    GATE CS 2014 Set 2, Q.14 (CS Q.4)

    If the matrix $A$ is such that $A=\left[\begin{array}{c}2\\ -4\\ 7\end{array}\right]\left[\begin{array}{ccc}1& 9& 5\end{array}\right]$ then the determinant of $A$ is equal to _____.

    Linear Algebra

44. GATE CS 2014 Set 2NAT2 marksMedium (estimate)

    GATE CS 2014 Set 2, Q.57 (CS Q.47)

    The product of the non-zero eigenvalues of the matrix…

    Linear Algebra

45. GATE CS 2014 Set 3MCQ1 markEasy (estimate)

    GATE CS 2014 Set 3, Q.14 (CS Q.4)

    Which one of the following statements is TRUE about every $n\times n$ matrix with only real eigenvalues?

    Linear Algebra

46. GATE CS 2014 Set 3NAT1 markEasy (estimate)

    GATE CS 2014 Set 3, Q.15 (CS Q.5)

    If $V_1$ and $V_2$ are 4-dimensional subspaces of a 6-dimensional vector space $V$, then the smallest possible dimension of $V_1\cap V_2$ is _____.

    Linear Algebra

47. GATE CS 2013MCQ1 markMedium (estimate)

    GATE CS 2013, Q.3 (CS Q.3)

    Which one of the following does NOT equal $\left|\begin{array}{ccc}1& x& x^2\\ 1& y& y^2\\ 1& z& z^2\end{array}\right|$?

    Linear Algebra

48. GATE CS 2012MCQ1 markMedium (estimate)

    GATE CS 2012, Q.11 (CS Q.11)

    Let $A$ be the $2\times 2$ matrix with elements $a_{11}=a_{12}=a_{21}=+1$ and $a_{22}=-1$. Then the eigenvalues of the matrix $A^{19}$ are

    Linear Algebra

49. GATE CS 2011MCQ2 marksEasy (estimate)

    GATE CS 2011, Q.40 (CS Q.40)

    Consider the matrix as given below. $\left[\begin{array}{ccc}1& 2& 3\\ 0& 4& 7\\ 0& 0& 3\end{array}\right]$ Which one of the following options provides the CORRECT values of the e...

    Linear Algebra

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• Propositional and First-Order Logic
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