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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 PYQ2016Engineering MathematicsLinear AlgebraMCQ · 1 mark
Let M4M^4= II,(where II denotes the identity matrix) and M≠IM \neq I, M2≠IM^2\neq I and M3≠IM^3\neq I. Then,for any natural number kk, M−1M^{-1} equals:
Official PYQ2016Engineering MathematicsLinear AlgebraNAT · 2 marks
A sequence x[n]x[n] is specified as [x[n]x[n–1]]=[1110]n[10]\begin{bmatrix}x[n] \\x[n – 1]\end{bmatrix}=\begin{bmatrix}1&1\\1&0\end{bmatrix}^n\begin{bmatrix}1\\0\end{bmatrix},for n≥2n \geq 2. The initial conditions are x[0]=1x[0] = 1, x[1]=1x[1] = 1, and x[n]=0x[n] = 0 for n<0n < 0. The value of x[12]x[12] is _________
Official PYQ2016Engineering MathematicsLinear AlgebraNAT · 1 mark
The value of xx for which the matrix A=[3249713−6−4−9+x]A= \begin{bmatrix} 3& 2 &4 \\ 9& 7 & 13\\ -6&-4 &-9+x \end{bmatrix} has zero as an eigenvalue is ________
Official PYQ2016Engineering MathematicsLinear AlgebraNAT · 2 marks
The matrix A=[a03725130024000b]A=\begin{bmatrix} a & 0 &3 &7 \\ 2& 5&1 &3 \\ 0& 0& 2 &4 \\ 0&0 & 0 &b \end{bmatrix} has det(A)=100\text{det}(A) = 100 and trace(A)=14\text{trace}(A) = 14. The value of ∣a−b∣\mid a-b \mid is ________
Official PYQ2016Engineering MathematicsLinear AlgebraMCQ · 1 mark
Consider a 2×22\times2 sqaure matrix A=[σxωσ],\textbf{A}= \begin{bmatrix} \sigma &x\\ \omega &\sigma \end{bmatrix}, where xx is unknown. If the eigen values of the matrix A\textbf{A} are (σ+jω)(\sigma + j\omega) and (σ−jω)(\sigma - j\omega), then xx is equal to
Official PYQ2016Engineering MathematicsLinear AlgebraMCQ · 2 marks
If the vectors e1=(1,0,2)e_1=(1,0,2), e2=(0,1,0)e_2=(0,1,0) and e3=(−2,0,1)e_3=(-2,0,1) form an orthogonal basis of the three-dimensional real space R3\mathbb{R}^3, then the vector u=(4,3,−3)∈R3\textbf{u}=(4,3,-3)\in \mathbb{R}^3 can be expressed as
Official PYQ2015Engineering MathematicsLinear AlgebraNAT · 1 mark
Consider a system of linear equations: x−2y+3z=−1,x−3y+4z=1, and −2x+4y−6z=k.x-2y+3z=-1, \\ x-3y+4z=1, \text{ and } \\ -2x+4y-6z=k. The value of kk for which the system has infinitely many solutions is ___________
Official PYQ2015Engineering MathematicsLinear AlgebraNAT · 1 mark
The value of pp such that the vector [123]\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} is an eigenvector of the matrix [412p2114−410]\begin{bmatrix} 4 & 1 & 2 \\ p & 2 & 1 \\ 14 & -4 & 10 \end{bmatrix} is _________.
Official PYQ2015Engineering MathematicsLinear AlgebraMCQ · 2 marks
Two sequences [a,b,c]\begin{bmatrix}a, & b, & c \end{bmatrix} and [A,B,C]\begin{bmatrix}A, & B, & C \end{bmatrix} are related as, …
Official PYQ2015Engineering MathematicsLinear AlgebraMCQ · 1 mark
The value of xx for which all the eigen-values of the matrix given below are real is [105+j4x20242−10]\begin{bmatrix} 10&5+j &4 \\ x&20 &2 \\4 &2 &-10 \end{bmatrix}
Official PYQ2015Engineering MathematicsLinear AlgebraMCQ · 1 mark
For A=[1tan⁡x−tan⁡x1],A = \begin{bmatrix} 1 &\tan x \\ -\tan x &1 \end{bmatrix}, the determinant of ATA−1A^{T}A^{-1} is
Official PYQ2014Engineering MathematicsLinear AlgebraMCQ · 1 mark
For matrices of same dimension M,NM, N and scalar cc, which one of these properties DOES NOT ALWAYS hold?
Official PYQ2014Engineering MathematicsLinear AlgebraNAT · 1 mark
A real (4×4)(4 \times 4) matrix AA satisfies the equation A2=IA^{2} = I, where II is the (4×4)(4 \times 4) identity matrix. The positive eigen value of AA is ______.
Official PYQ2014Engineering MathematicsLinear AlgebraNAT · 2 marks
Consider the matrix J6=[000001000010000100001000010000100000]J_{6} = \begin{bmatrix} 0&0 &0 &0 &0 &1 \\ 0& 0& 0& 0& 1&0 \\ 0& 0& 0& 1& 0&0 \\ 0&0 & 1& 0&0 &0 \\0 &1 &0 &0 &0 &0 \\1 &0 &0 &0 & 0& 0\end{bmatrix} which is obtained by reversing the order of the columns of the …
Official PYQ2014Engineering MathematicsLinear AlgebraMCQ · 2 marks
Consider the state space model of a system, as given below …
Official PYQ2014Engineering MathematicsLinear AlgebraNAT · 1 mark
The determinant of matrix AA is 55 and the determinant of matrix B is 4040. The determinant of matrix ABAB is ________
Official PYQ2014Engineering MathematicsLinear AlgebraMCQ · 2 marks
The system of linear equations (213301125)(abc)=(5−414)\begin{pmatrix} 2 & 1 & 3\\ 3&0 &1 \\ 1& 2 &5 \end{pmatrix} \begin{pmatrix} a\\ b\\ c \end{pmatrix} = \begin{pmatrix} 5\\ -4\\ 14 \end{pmatrix} has
Official PYQ2014Engineering MathematicsLinear AlgebraNAT · 2 marks
The maximum value of the determinant among all 2×22 \times 2 real symmetric matrices with trace 1414 is __________.
Official PYQ2014Engineering MathematicsLinear AlgebraMCQ · 1 mark
Match the application to appropriate numerical method. …
Official PYQ2014Engineering MathematicsLinear AlgebraMCQ · 2 marks
Which one of the following statements is NOT true for a square matrix AA?