Linear Algebra · Engineering Mathematics · GATE ECE
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GATE ECE 2024
Let $\mathbb{R}$ and $\mathbb{R}^3$ denote the set of real numbers and the three dimensional vector space over it, respectively. The value of $\alpha$...
GATE ECE 2023
Let the sets of eigenvalues and eigenvectors of a matrix B be $$\{ {\lambda _k}|1 \le k \le n\} $$ and $$\{ {v_k}|1 \le k \le n\} $$, respectively. Fo...
GATE ECE 2022
Consider a system of linear equations Ax = b, where
$$A = \left[ {\matrix{
1 \hfill & { - \sqrt 2 } \hfill & 3 \hfill \cr
{ - 1} \hfill & {\sq...
GATE ECE 2018
Let
M
be a real
4 $$ \times $$ 4
matrix. Consider the following statements :
S1:
M
has
4
linearly independent eigenvectors.
S2:
M
has
4
distinct eige...
GATE ECE 2018
Consider matrix $$A = \left[ {\matrix{
k & {2k} \cr
{{k^2} - k} & {{k^2}} \cr
} } \right]$$ and
vector $$X = \left[ {\matrix{
{...
GATE ECE 2017 Set 1
The rank of the matrix $$M = \left[ {\matrix{
5 & {10} & {10} \cr
1 & 0 & 2 \cr
3 & 6 & 6 \cr
} } \right]$$ i...
GATE ECE 2017 Set 1
Consider the $$5 \times 5$$ matrix $$A = \left[ {\matrix{
1 & 2 & 3 & 4 & 5 \cr
5 & 1 & 2 & 3 & 4 \cr
4 &...
GATE ECE 2016 Set 2
The value of $$x$$ for which the matrix $$A = \left[ {\matrix{
3 & 2 & 4 \cr
9 & 7 & {13} \cr
{ - 6} & { - 4} & {...
GATE ECE 2016 Set 1
Let $${M^4} = {\rm I}$$ (where $${\rm I}$$ denotes the identity matrix) and $$M \ne {\rm I},\,\,{M^2} \ne {\rm I}$$ and $${M^3} \ne {\rm I}$$. Then, ...
GATE ECE 2016 Set 3
Consider a $$2 \times 2$$ square matrix $$A = \left[ {\matrix{
\sigma & x \cr
\omega & \sigma \cr
} } \right]$$
Where $$x$$ is ...
GATE ECE 2015 Set 2
The value of $$'x'$$ for which all the eigenvalues of the matrix given below are real is $$\left[ {\matrix{
{10} & {5 + j} & 4 \cr
x ...
GATE ECE 2015 Set 1
Consider system of linear equations :
$$$x-2y+3z=-1$$$
$$$x-3y+4z=1$$$ and
$$$-2x+4y-6z=k,$$$
The value of $$'k'$$ for which the system has infinite...
GATE ECE 2015 Set 1
The value of $$'P'$$ such that the vector $$\left[ {\matrix{
1 \cr
2 \cr
3 \cr
} } \right]$$ is an eigenvector of the matrix $$\left...
GATE ECE 2015 Set 3
For $$A = \left[ {\matrix{
1 & {\tan x} \cr
{ - \tan x} & 1 \cr
} } \right],$$ the determinant of $${A^T}\,{A^{ - 1}}$$ is
GATE ECE 2014 Set 3
Which one of the following statements is NOT true for a square matrix $$A$$?
GATE ECE 2014 Set 2
The maximum value of the determinant among all $$2 \times 2$$ real symmetric matrices with trace $$14$$ is ______.
GATE ECE 2014 Set 2
The system of linear equations $$\left( {\matrix{
2 & 1 & 3 \cr
3 & 0 & 1 \cr
1 & 2 & 5 \cr
} } \right)\left(...
GATE ECE 2014 Set 2
The determinant of matrix $$A$$ is $$5$$ and the determinant of matrix $$B$$ is $$40.$$ The determinant of matrix $$AB$$ is _______.
GATE ECE 2014 Set 1
$$A$$ real $$\left( {4\,\, \times \,\,4} \right)$$ matrix $$A$$ satisfies the equation $${A^2} = {\rm I},$$ where $${\rm I}$$ is the $$\left( {4\,\, \...
GATE ECE 2014 Set 1
Consider the matrix $${J_6} = \left[ {\matrix{
0 & 0 & 0 & 0 & 0 & 1 \cr
0 & 0 & 0 & 0 & 1 & 0 \cr
...
GATE ECE 2014 Set 1
For matrices of same dimension $$M,N$$ and scalar $$c,$$ which one of these properties DOES NOT ALWAYS hold ?
GATE ECE 2013
The minimum eigenvalue of the following matrix is $$\left[ {\matrix{
3 & 5 & 2 \cr
5 & {12} & 7 \cr
2 & 7 & 5 \c...
GATE ECE 2013
Let $$A$$ be an $$m\,\, \times \,\,n$$ matrix and $$B$$ an $$n\,\, \times \,\,m$$ matrix. It is given that determinant $$\left( {{{\rm I}_m} + AB} \ri...
GATE ECE 2012
Given that $$A = \left[ {\matrix{
{ - 5} & { - 3} \cr
2 & 0 \cr
} } \right]$$ and $${\rm I} = \left[ {\matrix{
1 & 0 \cr
...
GATE ECE 2011
The system of equations $$x+y+z=6,$$ $$x+4y+6z=20,$$ $$x + 4y + \lambda z = \mu $$ has no solution for values of $$\lambda $$ and $$\mu $$ given by
GATE ECE 2010
The eigen values of a skew-symmetric matrix are
GATE ECE 2008
All the four entries of $$2$$ $$x$$ $$2$$ matrix
$$P = \left[ {\matrix{
{{p_{11}}} & {{p_{12}}} \cr
{{p_{21}}} & {{p_{22}}} \cr
}...
GATE ECE 2008
The system of linear equations $$\left. {\matrix{
{4x + 2y = 7} \cr
{2x + y = 6} \cr
} } \right\}$$ has
GATE ECE 2006
For the matrix $$\left[ {\matrix{
4 & 2 \cr
2 & 4 \cr
} } \right].$$ The eigen value corresponding to the eigen vector $$\left[ {\...
GATE ECE 2006
The rank of the matrix $$\left[ {\matrix{
1 & 1 & 1 \cr
1 & { - 1} & 0 \cr
1 & 1 & 1 \cr
} } \right]$$ is
GATE ECE 2000
The eigen values of the matrix $$\left[ {\matrix{
2 & { - 1} & 0 & 0 \cr
0 & 3 & 0 & 0 \cr
0 & 0 & { - 2}...
GATE ECE 1998
The eigen values of the matrix $$A = \left[ {\matrix{
0 & 1 \cr
1 & 0 \cr
} } \right]$$ are
GATE ECE 1994
The rank of $$\left( {m \times n} \right)$$ matrix $$\left( {m < n} \right)$$ cannot be more then
GATE ECE 1994
The following system of equations
$${{x_1} + {x_2} + {x_3} = 3}$$
$${{x_1} - {x_3} = 0}$$
$${{x_1} - {x_2} + {x_3} = 1}$$ has
Marks 2
GATE ECE 2024
Consider the matrix $\begin{bmatrix}1 & k \\ 2 & 1\end{bmatrix}$, where $k$ is a positive real number. Which of the following vectors is/are eigenvect...
GATE ECE 2023
Let $$x$$ be an $$n \times 1$$ real column vector with length $$l = \sqrt {{x^T}x} $$. The trace of the matrix $$P = x{x^T}$$ is
GATE ECE 2023
The state equation of a second order system is
$$x(t) = Ax(t),\,\,\,\,x(0)$$ is the initial condition.
Suppose $$\lambda_1$$ and $$\lambda_2$$ are two...
GATE ECE 2022
Let $$\alpha$$, $$\beta$$ two non-zero real numbers and v1, v2 be two non-zero real vectors of size 3 $$\times$$ 1. Suppose that v1 and v2 satisfy $$v...
GATE ECE 2017 Set 2
The rank of the matrix $$\left[ {\matrix{
1 & { - 1} & 0 & 0 & 0 \cr
0 & 0 & 1 & { - 1} & 0 \cr
0 & 1...
GATE ECE 2016 Set 2
The matrix $$A = \left[ {\matrix{
a & 0 & 3 & 7 \cr
2 & 5 & 1 & 3 \cr
0 & 0 & 2 & 4 \cr
0 & ...
GATE ECE 2016 Set 1
A sequence $$x\left[ n \right]$$ is specified as
$$$\left[ {\matrix{
{x\left[ n \right]} \cr
{x\left[ {n - 1} \right]} \cr
} } \right] = ...
GATE ECE 2016 Set 3
If the vectors $${e_1} = \left( {1,0,2} \right),\,{e_2} = \left( {0,1,0} \right)$$ and $${e_3} = \left( { - 2,0,1} \right)$$ form an orthogonal basis ...
GATE ECE 2009
The eigen values of the following matrix $$\left[ {\matrix{
{ - 1} & 3 & 5 \cr
{ - 3} & { - 1} & 6 \cr
0 & 0 & 3 ...
GATE ECE 2006
The eigen values and the correspondinng eigen vectors of a $$2 \times 2$$ matrix are given by
Eigen value
$${\lambda _1} = 8$$
$${\lambda _2} = 4$$
E...
GATE ECE 2005
Given the matrix $$\left[ {\matrix{
{ - 4} & 2 \cr
4 & 3 \cr
} } \right],$$ the eigen vector is
GATE ECE 2005
Given an orthogonal matrix $$A = \left[ {\matrix{
1 & 1 & 1 & 1 \cr
1 & 1 & { - 1} & { - 1} \cr
1 & { - 1} &a...
GATE ECE 2005
If $$A = \left[ {\matrix{
2 & { - 0.1} \cr
0 & 3 \cr
} } \right]$$ and $${A^{ - 1}} = \left[ {\matrix{
{{\raise0.5ex\hbox{$\scr...