GATE CE

Linear Algebra

Engineering Mathematics

(Past Years Questions)

Marks 1

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Consider the following simultaneous equations (with $${c_1}$$ and $${c_2}$$ being constants): $$$3{x_1} + 2{x_2} = {c_1...
GATE CE 2017 Set 2
The matrix $$P$$ is the inverse of a matrix $$Q.$$ If $${\rm I}$$ denotes the identity matrix, which one of the followin...
GATE CE 2017 Set 1
If the entries in each column of a square matrix $$M$$ add up to $$1$$, then an eigenvalue of $$M$$ is
GATE CE 2016 Set 1
Let $$A = \left[ {{a_{ij}}} \right],\,\,1 \le i,j \le n$$ with $$n \ge 3$$ and $${{a_{ij}} = i.j.}$$ The rank of $$A$$ i...
GATE CE 2015 Set 2
For what value of $$'p'$$ the following set of equations will have no solutions? $$$2x+3y=5$$$ $$$3x+py=10$$$
GATE CE 2015 Set 1
The determinant of matrix $$\left[ {\matrix{ 0 & 1 & 2 & 3 \cr 1 & 0 & 3 & 0 \cr 2 & 3 & 0 & 1 \cr 3 &...
GATE CE 2014 Set 2
The rank of the matrix $$\left[ {\matrix{ 6 & 0 & 4 & 4 \cr { - 2} & {14} & 8 & {18} \cr {14} & { - 14} & 0...
GATE CE 2014 Set 2
Given the matrices $$J = \left[ {\matrix{ 3 & 2 & 1 \cr 2 & 4 & 2 \cr 1 & 2 & 6 \cr } } \right]$$ and ...
GATE CE 2014 Set 1
The sum of Eigen values of the matrix, $$\left[ M \right]$$ is where $$\left[ M \right] = \left[ {\matrix{ {215} & {...
GATE CE 2014 Set 1
The eigen values of matrix $$\left[ {\matrix{ 9 & 5 \cr 5 & 8 \cr } } \right]$$ are
GATE CE 2012
A square matrix $$B$$ is symmetric if ____
GATE CE 2009
In the solution of the following set of linear equations by Gauss-elimination using partial pivoting $$$5x+y+2z=34,$$$ ...
GATE CE 2009
The product of matrices $${\left( {PQ} \right)^{ - 1}}P$$ is
GATE CE 2008
The eigenvalues of the matrix $$\left[ P \right] = \left[ {\matrix{ 4 & 5 \cr 2 & { - 5} \cr } } \right]$$ a...
GATE CE 2008
The following system of equations $$$x+y+z=3,$$$ $$$x+2y+3z=4,$$$ $$$x+4y+kz=6$$$ will not have a unique solution for $...
GATE CE 2008
Solution for the system defined by the set of equations $$4y+3z=8, 2x-z=2$$ & $$3x+2y=5$$ is
GATE CE 2006
Consider the following system of equations in three real variable $${x_1},$$ $${x_2}$$ and $${x_3}:$$ $$$2{x_1} - {x_2} ...
GATE CE 2005
Consider a non-homogeneous system of linear equations represents mathematically an over determined system. Such a system...
GATE CE 2005
Consider the matrices $$\,{X_{4x3,}}\,\,{Y_{4x3}}$$ $$\,\,{P_{2x3}}.$$ The order of $$\,{\left[ {P{{\left( {{X^T}Y} \ri...
GATE CE 2005
Consider the system of equations, $${A_{nxn}}\,\,{X_{nx1}}\,\, = \lambda \,{X_{nx1}}$$ where $$\lambda $$ is a scalar. L...
GATE CE 2005
Real matrices $$\,\,{\left[ A \right]_{3x1,}}$$ $$\,\,{\left[ B \right]_{3x3,}}$$ $$\,\,{\left[ C \right]_{3x5,}}$$ $$\...
GATE CE 2004
The eigen values of the matrix $$\left[ {\matrix{ 4 & { - 2} \cr { - 2} & 1 \cr } } \right]$$ are
GATE CE 2004
Given matrix $$\left[ A \right] = \left[ {\matrix{ 4 & 2 & 1 & 3 \cr 6 & 3 & 4 & 7 \cr 2 & 1 & 0 & 1 \cr ...
GATE CE 2003
Eigen values of the following matrix are $$\left[ {\matrix{ { - 1} & 4 \cr 4 & { - 1} \cr } } \right]$$
GATE CE 2002
The determinant of the following matrix $$\left[ {\matrix{ 5 & 3 & 2 \cr 1 & 2 & 6 \cr 3 & 5 & {10} \cr ...
GATE CE 2001
The eigen values of the matrix $$\left[ {\matrix{ 5 & 3 \cr 2 & 9 \cr } } \right]$$ are
GATE CE 2001
The product $$\left[ P \right]\,\,{\left[ Q \right]^T}$$ of the following two matrices $$\left[ P \right]\,$$ and $$\lef...
GATE CE 2001
If $$A,B,C$$ are square matrices of the same order then $${\left( {ABC} \right)^{ - 1}}$$ is equal be
GATE CE 2000
Consider the following two statements. $$(I)$$ The maximum number of linearly independent column vectors of a matrix $$...
GATE CE 2000
If $$A$$ is any $$nxn$$ matrix and $$k$$ is a scalar then $$\left| {kA} \right| = \alpha \left| A \right|$$ where $$\alp...
GATE CE 1999
The number of terms in the expansion of general determinant of order $$n$$ is
GATE CE 1999
The equation $$\left[ {\matrix{ 2 & 1 & 1 \cr 1 & 1 & { - 1} \cr y & {{x^2}} & x \cr } } \right] = 0$$ ...
GATE CE 1999
If $$A$$ is a real square matrix then $$A{A^T}$$ is
GATE CE 1998
In matrix algebra $$AS=AT$$ ($$A,S,T,$$ are matrices of appropriate order) implies $$S=T$$ only if
GATE CE 1998
The real symmetric matrix $$C$$ corresponding to the quadratic form $$Q = 4{x_1}{x_2} - 5{x_2}{x_2}$$ is
GATE CE 1998
Obtain the eigen values and eigen vectors of $$A = \left[ {\matrix{ 8 & -4 \cr 2 & { 2 } \cr } } \right].$$
GATE CE 1998
Inverse of matrix $$\left[ {\matrix{ 0 & 1 & 0 \cr 0 & 0 & 1 \cr 1 & 0 & 0 \cr } } \right]$$ is
GATE CE 1997
If $$A$$ and $$B$$ are two matrices and $$AB$$ exists then $$BA$$ exists,
GATE CE 1997
If the determinant of the matrix $$\left[ {\matrix{ 1 & 3 & 2 \cr 0 & 5 & { - 6} \cr 2 & 7 & 8 \cr } } ...
GATE CE 1997

Marks 2

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If $$A = \left[ {\matrix{ 1 & 5 \cr 6 & 2 \cr } } \right]\,\,and\,\,B = \left[ {\matrix{ 3 & 7 \cr 8...
GATE CE 2017 Set 2
Consider the matrix $$\left[ {\matrix{ 5 & { - 1} \cr 4 & 1 \cr } } \right].$$ Which one of the following st...
GATE CE 2017 Set 1
Consider the following linear system $$$x+2y-3z=a$$$ $$$2x+3y+3z=b$$$ $$$5x+9y-6z=c$$$ This system is consistent if $$a...
GATE CE 2016 Set 2
The two Eigen Values of the matrix $$\left[ {\matrix{ 2 & 1 \cr 1 & p \cr } } \right]$$ have a ratio of $$3:...
GATE CE 2015 Set 2
The smallest and largest Eigen values of the following matrix are : $$\left[ {\matrix{ 3 & { - 2} & 2 \cr 4 & {...
GATE CE 2015 Set 1
What is the minimum number of multiplications involved in computing the matrix product $$PQR?$$ Matrix $$P$$ has $$4$$ r...
GATE CE 2013
The inverse of the matrix $$\left[ {\matrix{ {3 + 2i} & i \cr { - i} & {3 - 2i} \cr } } \right]$$ is
GATE CE 2010
The inverse of $$2 \times 2$$ matrix $$\left[ {\matrix{ 1 & 2 \cr 5 & 7 \cr } } \right]$$ is
GATE CE 2007
For what values of $$\alpha $$ and $$\beta $$ the following simultaneous equations have an infinite number of solutions ...
GATE CE 2007
The minimum and maximum eigen values of matrix $$\left[ {\matrix{ 1 & 1 & 3 \cr 1 & 5 & 1 \cr 3 & 1 & 1 \c...
GATE CE 2007
For a given matrix $$A = \left[ {\matrix{ 2 & { - 2} & 3 \cr { - 2} & { - 1} & 6 \cr 1 & 2 & 0 \cr } } ...
GATE CE 2006

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