GATE EE

Linear Algebra

Engineering Mathematics

(Past Years Questions)

Marks 1

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The matrix $$A = \left[ {\matrix{ {{3 \over 2}} & 0 & {{1 \over 2}} \cr 0 & { - 1} & 0 \cr {{1 \over 2}} & ...
GATE EE 2017 Set 1
$$A$$ $$3 \times 3$$ matrix $$P$$ is such that , $${p^3} = P.$$ Then the eigen values of $$P$$ are
GATE EE 2016 Set 2
Consider $$3 \times 3$$ matrix with every element being equal to $$1.$$ Its only non-zero eigenvalue is __________.
GATE EE 2016 Set 1
We have a set of $$3$$ linear equations in $$3$$ unknown. $$'X \equiv Y'$$ means $$X$$ and $$Y$$ are equivalent statemen...
GATE EE 2015 Set 2
If the sum of the diagonal elements of a $$2 \times 2$$ matrix is $$-6$$, then the maximum possible value of determinant...
GATE EE 2015 Set 1
Which one of the following statements is true for all real symmetric matrices?
GATE EE 2014 Set 2
Given a system of equations $$$x + 2y + 2z = {b_1}$$$ $$$5x + y + 3z = {b_2}$$$ Which of the following is true its solu...
GATE EE 2014 Set 1
Given that $$A = \left[ {\matrix{ { - 5} & { - 3} \cr 2 & 0 \cr } } \right]$$ and $${\rm I} = \left[ {\matri...
GATE EE 2012
An eigen vector of $$p = \left[ {\matrix{ 1 & 1 & 0 \cr 0 & 2 & 2 \cr 0 & 0 & 3 \cr } } \right]$$ is
GATE EE 2010
The trace and determinant of a $$2 \times 2$$ matrix are shown to be $$-2$$ and $$-35$$ respectively. Its eigen values a...
GATE EE 2009
The characteristic equation of a $$3\,\, \times \,\,3$$ matrix $$P$$ is defined as $$\alpha \left( \lambda \right) = \...
GATE EE 2008
$$A$$ is $$m$$ $$x$$ $$n$$ full rank matrix with $$m > n$$ and $${\rm I}$$ is an identity matrix. Let matrix $${A^ + } =...
GATE EE 2008
$$X = {\left[ {\matrix{ {{x_1}} & {{x_2}} & {.......\,{x_n}} \cr } } \right]^T}$$ is an $$n$$-tuple non- zero vec...
GATE EE 2007
In the matrix equation $$PX=Q$$ which of the following is a necessary condition for the existence of atleast one solutio...
GATE EE 2005
The determinant of the matrix $$\left[ {\matrix{ 1 & 0 & 0 & 0 \cr {100} & 1 & 0 & 0 \cr {100} & {200} & 1 ...
GATE EE 2002
If $$A = \left[ {\matrix{ 1 & { - 2} & { - 1} \cr 2 & 3 & 1 \cr 0 & 5 & { - 2} \cr } } \right]$$ and $$...
GATE EE 1999
Find the eigen values and eigen vectors of the matrix $$\left[ {\matrix{ 3 & { - 1} \cr { - 1} & 3 \cr } } \...
GATE EE 1999
If the vector $$\left[ {\matrix{ 1 \cr 2 \cr { - 1} \cr } } \right]$$ is an eigen vector of $$A = \left...
GATE EE 1998
$$A = \left[ {\matrix{ 2 & 0 & 0 & { - 1} \cr 0 & 1 & 0 & 0 \cr 0 & 0 & 3 & 0 \cr { - 1} & 0 & 0 & 4 ...
GATE EE 1998
If $$A = \left[ {\matrix{ 5 & 0 & 2 \cr 0 & 3 & 0 \cr 2 & 0 & 1 \cr } } \right]$$ then $${A^{ - 1}} = $...
GATE EE 1998
A set of linear equations is represented by the matrix equations $$Ax=b.$$ The necessary condition for the existence of ...
GATE EE 1998
Express the given matrix $$A = \left[ {\matrix{ 2 & 1 & 5 \cr 4 & 8 & {13} \cr 6 & {27} & {31} \cr } } ...
GATE EE 1997
The inverse of the matrix $$S = \left[ {\matrix{ 1 & { - 1} & 0 \cr 1 & 1 & 1 \cr 0 & 0 & 1 \cr } } \ri...
GATE EE 1995
Given the matrix $$A = \left[ {\matrix{ 0 & 1 & 0 \cr 0 & 0 & 1 \cr { - 6} & { - 11} & { - 6} \cr } } \...
GATE EE 1995
The rank of the following $$(n+1)$$ $$x$$ $$(n+1)$$ matrix, where $$'a'$$ is a real number is $$$\left[ {\matrix{ 1 ...
GATE EE 1995
$$A$$ $$\,\,5 \times 7$$ matrix has all its entries equal to $$1.$$ Then the rank of a matrix is
GATE EE 1994
The eigen values of the matrix $$\left[ {\matrix{ a & 1 \cr a & 1 \cr } } \right]$$ are
GATE EE 1994
The number of linearly independent solutions of the system of equations $$\left[ {\matrix{ 1 & 0 & 2 \cr 1 & { ...
GATE EE 1994

Marks 2

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The eigen values of the matrix given below are $$\left[ {\matrix{ 0 & 1 & 0 \cr 0 & 0 & 1 \cr 0 & { - 3} & ...
GATE EE 2017 Set 2
Let $$P = \left[ {\matrix{ 3 & 1 \cr 1 & 3 \cr } } \right].$$ Consider the set $$S$$ of all vectors $$\left(...
GATE EE 2016 Set 2
Let the eigenvalues of a $$2 \times 2$$ matrix $$A$$ be $$1,-2$$ with eigenvectors $${x_1}$$ and $${x_2}$$ respectively....
GATE EE 2016 Set 1
Let $$A$$ be a $$4 \times 3$$ real matrix which rank$$2.$$ Which one of the following statement is TRUE?
GATE EE 2016 Set 1
The maximum value of $$'a'$$ such that the matrix $$\left[ {\matrix{ { - 3} & 0 & { - 2} \cr 1 & { - 1} & 0 \cr...
GATE EE 2015 Set 1
$$A = \left[ {\matrix{ p & q \cr r & s \cr } } \right];B = \left[ {\matrix{ {{p^2} + {q^2}} & {pr + qs} ...
GATE EE 2014 Set 3
A system matrix is given as follows $$$A = \left[ {\matrix{ 0 & 1 & { - 1} \cr { - 6} & { - 11} & 6 \cr { ...
GATE EE 2014 Set 1
The equation $$\left[ {\matrix{ 2 & { - 2} \cr 1 & { - 1} \cr } } \right]\left[ {\matrix{ {{x_1}} \cr ...
GATE EE 2013
A matrix has eigen values $$-1$$ and $$-2.$$ The corresponding eigenvectors are $$\left[ {\matrix{ 1 \cr { - 1} ...
GATE EE 2013
The two vectors $$\left[ {\matrix{ 1 & 1 & 1 \cr } } \right]$$ and $$\left[ {\matrix{ 1 & a & {{a^2}} \cr ...
GATE EE 2011
The matrix $$\left[ A \right] = \left[ {\matrix{ 2 & 1 \cr 4 & { - 1} \cr } } \right]$$ is decomposed into a...
GATE EE 2011
For the set of equations $$${x_1} + 2{x_2} + {x_3} + 4{x_4} = 2,$$$ $$$3{x_1} + 6{x_2} + 3{x_3} + 12{x_4} = 6.$$$ The f...
GATE EE 2010
If the rank of a $$5x6$$ matrix $$Q$$ is $$4$$ then which one of the following statements is correct?
GATE EE 2008
Let $$P$$ be $$2x2$$ real orthogonal matrix and $$\overline x $$ is a real vector $${\left[ {\matrix{ {{x_1}} & {{x_2...
GATE EE 2008
$${q_1},\,{q_2},{q_3},.......{q_m}$$ are $$n$$-dimensional vectors with $$m ...
GATE EE 2007
If $$A = \left[ {\matrix{ { - 3} & 2 \cr { - 1} & 0 \cr } } \right]$$ then $$A$$ satisfies the relation
GATE EE 2007
If $$A = \left[ {\matrix{ { - 3} & 2 \cr { - 1} & 0 \cr } } \right]\,$$ then $${A^9}$$ equals
GATE EE 2007
Let $$x$$ and $$y$$ be two vectors in a $$3-$$ dimensional space and $$ $$ denote their dot product. Then the determina...
GATE EE 2007
For the matrix $$P = \left[ {\matrix{ 3 & { - 2} & 2 \cr 0 & { - 2} & 1 \cr 0 & 0 & 1 \cr } } \right],$...
GATE EE 2005
If $$R = \left[ {\matrix{ 1 & 0 & { - 1} \cr 2 & 1 & { - 1} \cr 2 & 3 & 2 \cr } } \right]$$ then the t...
GATE EE 2005

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