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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 PYQ2014Engineering MathematicsLinear AlgebraMCQ · 2 marks
The state equation of a second-order linear system is given by x˙(t)=Ax(t),        x(0)=x0\dot{x}(t)=Ax(t), \:\:\:\:\:\:\:\:x(0)=x_{0} For x0=[1−1],x_{0}= \begin{bmatrix} 1\\ -1 \end{bmatrix}, x(t)=[e−t−e−t],x(t)= \begin{bmatrix} e^{-t}\\ -e^{-t} \end{bmatrix}, and for x0=[01],x_{0}= \begin{bmatrix} 0\\ 1 \end{bmatrix}, x(t)[e−t−e−2t−e−t+2e−2t]x(t) \begin{bmatrix} e^{-t}-e^{-2t}\\ -e^{-t}+2e^{-2t} \end{bmatrix}. When …
Official PYQ2014Engineering MathematicsLinear AlgebraMCQ · 2 marks
The state transition matrix ϕ(t)\phi(t) of a system [x1x2]=[0100][x1x2]\begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} is
Official PYQ2013Engineering MathematicsLinear AlgebraMCQ · 1 mark
The minimum eigenvalue of the following matrix is [3525127275]\begin{bmatrix} 3& 5& 2\\5 &12 &7 \\2 &7 & 5\end{bmatrix}
Official PYQ2013Engineering MathematicsLinear AlgebraMCQ · 2 marks
Let AA be an m×nm \times n matrix and BB an n×mn \times m matrix. It is given that determinant (Im+AB)=(I_{m} + AB) = determinant (In+BA),(I_{n} + BA), where IkI_{k} is the k×kk \times k identity matrix. Using the above property, the determinant of the matrix given below is …
Official PYQ2013Engineering MathematicsLinear AlgebraMCQ · 2 marks
The DFT\text{DFT} of a vector [abcd]\begin{bmatrix} a & b & c & d \end{bmatrix} is the vector [αβγδ].\begin{bmatrix} \alpha & \beta & \gamma & \delta \end{bmatrix}. consider the product …
Official PYQ2012Engineering MathematicsLinear AlgebraMCQ · 2 marks
Given that A=[−5−320]A=\begin{bmatrix} -5 &-3 \\ 2 &0\end{bmatrix} and I=[1001]I=\begin{bmatrix} 1 & 0 \\ 0 &1\end{bmatrix}, the value of A3A^3 is
Official PYQ2009Engineering MathematicsLinear AlgebraMCQ · 1 mark
The eigen values of the following matrix are [−135−3−16003]\left[\begin{array}{ccc} -1 & 3 & 5 \\ -3 & -1 & 6 \\ 0 & 0 & 3 \end{array}\right]
Official PYQ2008Engineering MathematicsLinear AlgebraMCQ · 1 mark
All the four entries of the 2×22 \times 2 matrix P=[p11p12p21p22]\mathbf{P}=\left[\begin{array}{ll}p_{11} & p_{12} \\ p_{21} & p_{22}\end{array}\right] are nonzero, and one of its eigenvalues is zero. Which of the following statements is true?
Official PYQ2008Engineering MathematicsLinear AlgebraMCQ · 1 mark
The system of linear equations 4x+2y=72x+y=6\begin{array}{l} 4 x+2 y=7 \\ 2 x+y=6 \end{array} has
Official PYQ2008Engineering MathematicsLinear AlgebraMCQ · 1 mark
Consider the matrix P=[01−2−3]\mathbf{P}=\left[\begin{array}{cc}0 & 1 \\ -2 & -3\end{array}\right]. The value of eP\mathbf{e}^{\mathbf{P}} is
Official PYQ2007Engineering MathematicsLinear AlgebraMCQ · 1 mark
It is given that X1,X2,⋯ ,XMX_{1}, X_{2}, \cdots ,X_{M} are MM non-zero, orthogonal vectors. The dimension of the vector space spanned by the 2M2 M vectors X1,X2,⋯ ,XM,−X1,−X2,⋯ ,−XMX_{1}, X_{2}, \cdots, X_{M},-X_{1},-X_{2}, \cdots,-X_{M} is
Official PYQ2006Engineering MathematicsLinear AlgebraMCQ · 1 mark
The rank of the matrix [1111−10111]\left[\begin{array}{ccc}1 & 1 & 1 \\ 1 & -1 & 0 \\ 1 & 1 & 1\end{array}\right] is
Official PYQ2006Engineering MathematicsLinear AlgebraMCQ · 1 mark
The eigenvalues and the corresponding eigen vectors of a 2×22 \times 2 matrix are given by EigenvalueEigenvectorλ1=8v1=[11]λ2=4v2=[1–1]\begin{array}{cc} \textbf{Eigenvalue}& \textbf{Eigenvector} \\ \lambda_1=8 & \text{v}_{1} = \left[ \begin{array} {} 1 \\ 1 \end{array} \right] \\ \lambda_2=4 & \text{v}_{2} = \left[ \begin{array} {c} 1 \\ – 1 \end{array} \right] \end{array} The matrix …
Official PYQ2006Engineering MathematicsLinear AlgebraMCQ · 1 mark
For the matrix [4224]\left[\begin{array}{ll}4 & 2 \\ 2 & 4\end{array}\right], the eigen value corresponding to the eigenvector [101101]\left[\begin{array}{l}101 \\ 101\end{array}\right] is
Official PYQ2005Engineering MathematicsLinear AlgebraMCQ · 1 mark
Given the matrix [−4243]\left[\begin{array}{cc}-4 & 2 \\ 4 & 3\end{array}\right], the eigenvector is
Official PYQ2005Engineering MathematicsLinear AlgebraMCQ · 1 mark
Let, A=[2−0.103]\mathrm{A}=\left[\begin{array}{cc}2 & -0.1 \\ 0 & 3\end{array}\right] and A−1=[12a0 b]\mathrm{A}^{-1}=\left[\begin{array}{ll}\frac{1}{2} & \mathrm{a} \\ 0 & \mathrm{~b}\end{array}\right] Then (a+b)=(a+b)=
Official PYQ2005Engineering MathematicsLinear AlgebraMCQ · 1 mark
Match the following and choose the correct combination: …
Official PYQ2005Engineering MathematicsLinear AlgebraMCQ · 1 mark
Given an orthogonal matrix A=[111111−1−11−100001−1]A=\left[\begin{array}{cccc}1 & 1 & 1 & 1 \\ 1 & 1 & -1 & -1 \\ 1 & -1 & 0 & 0 \\ 0 & 0 & 1 & -1\end{array}\right] [AAT]−1\left[\mathrm{AA}^{\mathrm{T}}\right]^{-1} is
Official PYQ2004Engineering MathematicsLinear AlgebraMCQ · 1 mark
If A=[−221−3]A=\left[\begin{array}{cc}-2 & 2 \\ 1 & -3\end{array}\right], then sin⁡At\sin A t is
Official PYQ2000Engineering MathematicsLinear AlgebraMCQ · 1 mark
The eigen values of the matrix [2−100030000−2000−14]\left[\begin{array}{rrrr}2 & -1 & 0 & 0 \\ 0 & 3 & 0 & 0 \\ 0 & 0 & -2 & 0 \\ 0 & 0 & -1 & 4\end{array}\right] are