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GATE EC Linear Algebra questions

Practise Linear Algebra questions for GATE EC. Explore official PYQs and original topic-wise quizzes, check answers and view available solutions. Free, no sign-in.

Start with matrix dimensions and the conditions for each operation. Check rank, eigenvalue and invertibility assumptions before applying a theorem.

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20 on this page
Official PYQ2026Engineering MathematicsLinear AlgebraMSQ · 1 mark
Consider the matrix M=[211130−1ab]\mathrm{M}=\left[\begin{array}{ccc}2 & 1 & 1 \\ 1 & 3 & 0 \\ -1 & a & b\end{array}\right]. Which of the following options is/ are TRUE if det⁡(M)≠0?\operatorname{det}(\mathrm{M}) \neq 0 ?
Official PYQ2025Engineering MathematicsLinear AlgebraMCQ · 1 mark
Consider the matrix AA below: A=[2345067800αβ000γ]A=\left[\begin{array}{llll} 2 & 3 & 4 & 5 \\ 0 & 6 & 7 & 8 \\ 0 & 0 & \alpha & \beta \\ 0 & 0 & 0 & \gamma \end{array}\right] For which of the following combinations of α,β\alpha, \beta, and γ\gamma, is the rank of AA at least three? Only (i), (iii), and (iv) Only …
Official PYQ2024Engineering MathematicsLinear AlgebraNAT · 1 mark
Let R\mathbb{R} and R3\mathbb{R}^{3} denote the set of real numbers and the three dimensional vector space over it, respectively. The value of α\alpha for which the set of vectors …
Official PYQ2024Engineering MathematicsLinear AlgebraMSQ · 2 marks
​​​​​Consider the matrix [1k21]\left[\begin{array}{ll}1 & k \\ 2 & 1\end{array}\right], where kk is a positive real number. Which of the following vectors is/are eigenvector(s) of this matrix?
Official PYQ2023Engineering MathematicsLinear AlgebraMCQ · 1 mark
Let v1=[120]v_1=\left[\begin{array}{l}1 \\ 2 \\ 0\end{array}\right] and v2=[213]v_2=\left[\begin{array}{l}2 \\ 1 \\ 3\end{array}\right] be two vectors. The value of the coefficient α\alpha in the expression v1=αv2+ev_1=\alpha v_2+e, which minimizes the length of the error vector ee, is
Official PYQ2023Engineering MathematicsLinear AlgebraMCQ · 1 mark
Let the sets of eigenvalues and eigenvectors of a matrix BB be {λk∣1≤k≤n}\left\{\lambda_k \mid 1 \leq k \leq n\right\} and {vk∣1≤k≤n}\left\{v_k \mid 1 \leq k \leq n\right\}, respectively. For any invertible matrix PP, the sets of eigenvalues and eigenvectors of the matrix AA, where B=P−1APB=P^{-1} A P, respectively, are
Official PYQ2023Engineering MathematicsLinear AlgebraMCQ · 2 marks
Let x\boldsymbol{x} be an n×1n \times 1 real column vector with length l=xTxl=\sqrt{\boldsymbol{x}^{T} \boldsymbol{x}} The trace of the matrix P=xxTP=\boldsymbol{x} \boldsymbol{x}^{T} is
Official PYQ2022Engineering MathematicsLinear AlgebraMCQ · 1 mark
Consider a system of linear equations Ax=b,Ax = b, where A=[1–23–12–3],b=[13].A =\begin{bmatrix} 1 & – \sqrt{2} & 3 \\ – 1 & \sqrt{2} & – 3 \end{bmatrix}, \quad b = \begin{bmatrix} 1 \\ 3 \end{bmatrix}. This system of equations admits ______________.
Official PYQ2022Engineering MathematicsLinear AlgebraMCQ · 2 marks
Let α,β\alpha, \beta be two non-zero real numbers and v1,v2v_{1}, v_{2} be two non-zero real vectors of size 3×1.3 \times 1. Suppose that v1v_{1} and v2v_{2} satisfy v1Tv2=0,v1Tv1=1,v_{1}^{T} v_{2} = 0, v_{1}^{T} v_{1} = 1, and v2Tv2=1.v_{2}^{T} v_{2} = 1. Let AA be the 3×33 \times 3 matrix given by: A=αv1v1T+βv2v2TA = \alpha v_{1} v_{1}^{T} + \beta v_{2} v_{2}^{T} The …
Official PYQ2021Engineering MathematicsLinear AlgebraNAT · 1 mark
If the vectors (1.0, −1.0, 2.0)(1.0,\:-1.0,\:2.0), (7.0, 3.0, x)(7.0,\:3.0,\:x) and (2.0, 3.0, 1.0)(2.0,\:3.0,\:1.0) in R3\mathbb{R}^{3} are linearly dependent, the value of xx is __________
Official PYQ2021Engineering MathematicsLinear AlgebraNAT · 2 marks
A real 2×22\times2 non-singular matrix AA with repeated eigenvalue is given as A=[x−3.03.04.0]A=\begin{bmatrix} x & -3.0\\ 3.0 & 4.0 \end{bmatrix} where xx is a real positive number. The value of xx (rounded off to one decimal place) is ________________
Official PYQ2020Engineering MathematicsLinear AlgebraMCQ · 1 mark
If v1,v2,…,v6v_{1},v_{2}, \dots ,v_{6} are six vectors in R4\mathbb{R}^{4} , which one of the following statements is FALSE\text{FALSE}?
Official PYQ2020Engineering MathematicsLinear AlgebraMCQ · 2 marks
Consider the following system of linear equations. x1+2x2=b1;2x1+4x2=b2;3x1+7x2=b3;3x1+9x2=b4\begin{array}{llll} x_{1}+2x_{2}=b_{1} ; & 2x_{1}+4x_{2}=b_{2}; & 3x_{1}+7x_{2}=b_{3} ; & 3x_{1}+9x_{2}=b_{4} \end{array} Which one of the following conditions ensures that a solution exists for the above system?
Official PYQ2019Engineering MathematicsLinear AlgebraNAT · 1 mark
The number of distinct eigenvalues of the matrix A=[2233011100330002]A=\begin{bmatrix} 2&2&3&3\\0&1&1&1\\0&0&3&3\\0&0&0&2 \end{bmatrix} is equal to ____________.
Official PYQ2018Engineering MathematicsLinear AlgebraMCQ · 1 mark
Let M\text{M} be a real 4×44\times 4 matrix. Consider the following statements: S1:MS1: M has 44 linearly independent eigenvectors. S2:MS2: M has 44 distinct eigenvalues. S3:MS3: M is non-singular (invertible). Which one among the following is TRUE?
Official PYQ2018Engineering MathematicsLinear AlgebraNAT · 1 mark
Consider matrix A=[k2kk2−kk2]A=\begin{bmatrix} k & 2k\\ k^{2}-k & k^{2} \end{bmatrix} and vector x=[x1x2].x=\begin{bmatrix} x_{1}\\ x_{2} \end{bmatrix}. The number of distinct real value of kk for which the equation Ax=0Ax=0 has infinitely many solutions is _________.
Official PYQ2017Engineering MathematicsLinear AlgebraMCQ · 1 mark
Consider the 5 ×\times 5 matrix [1234551234451233451223451]\begin{bmatrix} 1&2&3&4&5\\ 5&1 &2& 3 &4\\ 4&5&1&2&3\\ 3&4&5&1&2\\ 2&3&4&5&1 \end{bmatrix} It is given that A has only one real eigenvalue. Then the real eigenvalue of A is
Official PYQ2017Engineering MathematicsLinear AlgebraMCQ · 1 mark
The rank of the matrix M=[51010102366]\textbf{M} = \begin{bmatrix} 5&10&10 \\ 1 &0 &2 \\ 3&6&6 \end{bmatrix} is
Official PYQ2017Engineering MathematicsLinear AlgebraMCQ · 1 mark
Consider the following statements about the linear dependence of the real valued function y1=1,y2=xy_{1}=1,y_{2}=x and y3=x2y_{3}=x^{2} over the field of real numbers. Which one among the following is correct? Both I and II are true Both I and III are true Both II and IV are true Both III and IV are true
Official PYQ2017Engineering MathematicsLinear AlgebraNAT · 1 mark
The rank of the matrix [1−1000001−1001−100−100010001−1]\begin{bmatrix} 1 & -1& 0& 0& 0& \\ 0& 0& 1& -1& 0& \\ 0& 1& -1& 0& 0& \\ -1& 0& 0& 0& 1& \\ 0& 0& 0& 1& -1& \end{bmatrix} is ________.