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GATE CSE
Linear Algebra
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Previous Years Questions
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Marks 1
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Which of the following is/are the eigen vector(s) for the matrix given below? $$\left( {\matrix{ { - 9} & { - 6} & { ...
GATE CSE 2022
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Consider solving the following system of simultaneous equations using LU decomposition. x1 + x2 $$-$$ 2x3 = 4 x1 + 3x2 $...
GATE CSE 2022
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Which one of the following is the closed form for the generating function of the sequence (an}n $$\ge$$ 0 defined below?...
GATE CSE 2022
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Consider the following two statements with respect to the matrices Am $$\times$$ n , Bn $$\times$$ m , Cn$$\times$$ n a...
GATE CSE 2022
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Let X be a square matrix. Consider the following two statements on X. I. X is invertible. II. Determinant of X is non-z...
GATE CSE 2019
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Consider a matrix $$A = u{v^T}$$ where $$u = \left( {\matrix{ 1 \cr 2 \cr } } \right),v = \left( {\matrix{ ...
GATE CSE 2018
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Let $${c_1},.....,\,\,{c_n}$$ be scalars, not all zero, such that $$\sum\limits_{i = 1}^n {{c_i}{a_i} = 0} $$ where $${{...
GATE CSE 2017 Set 1
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Let $$P = \left[ {\matrix{ 1 & 1 & { - 1} \cr 2 & { - 3} & 4 \cr 3 & { - 2} & 3 \c...
GATE CSE 2017 Set 2
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Let $${a_n}$$ be the number of $$n$$-bit strings that do NOT contain two consecutive $$1s.$$ Which one of the following ...
GATE CSE 2016 Set 1
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Suppose that the eigen values of matrix $$A$$ are $$1, 2, 4.$$ The determinant of $${\left( {{A^{ - 1}}} \right)^T}$$ is...
GATE CSE 2016 Set 2
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Consider the system, each consisting of m linear equations in $$n$$ variables. $$I.$$ $$\,\,\,$$ If $$m < n,$$ then ...
GATE CSE 2016 Set 2
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Two eigenvalues of a $$3 \times 3$$ real matrix $$P$$ are $$\left( {2 + \sqrt { - 1} } \right)$$ and $$3.$$ The determin...
GATE CSE 2016 Set 1
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The number of divisors of $$2100$$ is ___________.
GATE CSE 2015 Set 2
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In the LU decomposition of the matrix $$\left[ {\matrix{ 2 & 2 \cr 4 & 9 \cr } } \right]$$, if the d...
GATE CSE 2015 Set 1
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The larger of the two eigenvalues of the matrix $$\left[ {\matrix{ 4 & 5 \cr 2 & 1 \cr } } \right]$$...
GATE CSE 2015 Set 2
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In the given matrix $$\left[ {\matrix{ 1 & { - 1} & 2 \cr 0 & 1 & 0 \cr 1 & 2 & 1 ...
GATE CSE 2015 Set 3
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If $${V_1}$$ and $${V_2}$$ are 4-dimensional subspaces of a 6-dimensional vector space V, then the smallest possible dim...
GATE CSE 2014 Set 3
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Which one of the following statements is TRUE about every $$n\,\, \times \,n$$ matrix with only real eigen values?
GATE CSE 2014 Set 3
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Consider the following system of equations: 3x + 2y = 1 4x + 7z = 1 x + y + z =3 x - 2y + 7z = 0 The number of solu...
GATE CSE 2014 Set 1
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If the matrix A is such that $$$A = \left[ {\matrix{ 2 \cr { - 4} \cr 7 \cr } } \right]\,\,\left[ {\ma...
GATE CSE 2014 Set 2
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The value of the dot product of the eigenvectors corresponding to any pair of different eigen values of a 4-by-4 symmetr...
GATE CSE 2014 Set 1
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Which of the following does not equal $$\left| {\matrix{ 1 & x & {{x^2}} \cr 1 & y & {{y^2}} \...
GATE CSE 2013
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Let $$A$$ be the $$2 \times 2$$ matrix with elements $${a_{11}} = {a_{12}} = {a_{21}} = + 1$$ and $${a_{22}} = - 1$$. ...
GATE CSE 2012
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Consider the following matrix $$A = \left[ {\matrix{ 2 & 3 \cr x & y \cr } } \right].$$ If the eige...
GATE CSE 2010
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The following system of equations $${x_1}\, + \,{x_2}\, + 2{x_3}\, = 1$$ $${x_1}\, + \,2 {x_2}\, + 3{x_3}\, = 2$$ $${...
GATE CSE 2008
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Let $$A$$ be the matrix $$\left[ {\matrix{ 3 & 1 \cr 1 & 2 \cr } } \right]$$. What is the maximum v...
GATE CSE 2007
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The determination of the matrix given below is $$$\left[ {\matrix{ 0 & 1 & 0 & 2 \cr { - 1} & 1 ...
GATE CSE 2005
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The number of different $$n \times n$$ symmetric matrices with each elements being either $$0$$ or $$1$$ is
GATE CSE 2004
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What values of x, y and z satisfy the following system of linear equations? $$$\left[ {\matrix{ 1 & 2 & 3 \c...
GATE CSE 2004
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Let A, B, C, D be $$n\,\, \times \,\,n$$ matrices, each with non-zero determination. If ABCD = I, then $${B^{ - 1}}$$ is...
GATE CSE 2004
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$$A$$ system of equations represented by $$AX=0$$ where $$X$$ is a column vector of unknown and $$A$$ is a square matrix...
GATE CSE 2003
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The rank of the matrix$$\left[ {\matrix{ 1 & 1 \cr 0 & 0 \cr } } \right]\,\,is$$
GATE CSE 2002
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Consider the following statements: S1: The sum of two singular n x n matrices may be non-singular S2: The sum of two n...
GATE CSE 2001
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An $$n\,\, \times \,\,n$$ array v is defined as follows v[i, j] = i - j for all i, j, $$1\,\, \le \,\,i\,\, \le \,\,n,\,...
GATE CSE 2000
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The determinant of the matrix $$$\left[ {\matrix{ 2 & 0 & 0 & 0 \cr 8 & 1 & 7 & 2 \cr ...
GATE CSE 2000
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Consider the following set a equations x + 2y = 5 4x + 8y = 12 3x + 6y + 3z = 15 This set
GATE CSE 1998
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The determination of the matrix $$$\left[ {\matrix{ 6 & { - 8} & 1 & 1 \cr 0 & 2 & 4 & 6...
GATE CSE 1997
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Let $$A = \left[ {\matrix{ {{a_{11}}} & {{a_{12}}} \cr {{a_{21}}} & {{a_{22}}} \cr } } \right]\,\,$$...
GATE CSE 1996
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Let AX = B be a system of linear equations where A is an m x n matrix and B is a $$m\,\, \times \,\,1$$ column vector an...
GATE CSE 1996
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The rank of the following (n + 1) x (n + 1) matrix, where a is a real number is $$$\left[ {\matrix{ 1 & a & ...
GATE CSE 1995
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The rank of the following (n + 1) x (n + 1) matrix, where a is a real number is $$$\left[ {\matrix{ 1 & a & ...
GATE CSE 1995
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The rank of the matrix $$\left[ {\matrix{ 0 & 0 & { - 3} \cr 9 & 3 & 5 \cr 3 & 1 & ...
GATE CSE 1994
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The inverse of the matrix $$\left[ {\matrix{ 1 & 0 & 1 \cr { - 1} & 1 & 1 \cr 0 & 1 &am...
GATE CSE 1994
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If $$A = \left[ {\matrix{ 1 & 0 & 0 & 1 \cr 0 & { - 1} & 0 & { - 1} \cr 0 & 0 &...
GATE CSE 1993
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The eigen vector (s) of the matrix $$\left[ {\matrix{ 0 & 0 & \alpha \cr 0 & 0 & 0 \cr 0 ...
GATE CSE 1993
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Marks 2
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Consider the following matrix. $$\left( {\begin{array}{*{20}{c}} 0&1&1&1\\ 1&0&1&1\\ 1&1&am...
GATE CSE 2021 Set 1
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Let A and B be two n$$ \times $$n matrices over real numbers. Let rank(M) and det(M) denote the rank and determinant of ...
GATE CSE 2020
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Consider a matrix P whose only eigenvectors are the multiples of $$\left[ {\matrix{ 1 \cr 4 \cr } } \right]...
GATE CSE 2018
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Which one of the following is a closed form expression for the generating function of the sequence $$\left\{ {{a_n}} \ri...
GATE CSE 2018
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Let $$A$$ be $$n\,\, \times \,\,n$$ real valued square symmetric matrix of rank $$2$$ with $$\sum\limits_{i = 1}^n {\sum...
GATE CSE 2017 Set 1
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If the characteristic polynomial of a $$3 \times 3$$ matrix $$M$$ over $$R$$(the set of real numbers) is $${\lambda ^3} ...
GATE CSE 2017 Set 2
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The value of the expression $${13^{99}}$$ ($$mod$$ $$17$$), in the range $$0$$ to $$16,$$ is ______________ .
GATE CSE 2016 Set 2
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Consider the recurrence relation $${a_1} = 8,\,{a_n} = 6{n^2} + 2n + {a_{n - 1}}.$$ Let $${a_{99}} = K \times {10^4}.$$...
GATE CSE 2016 Set 1
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Let $${A_1},\,{A_2},\,{A_3}$$ and $${A_4}$$ be four matrices of dimensions $$10 \times 5,\,5 \times 20,\,20 \times 10,$$...
GATE CSE 2016 Set 2
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Let $${a_n}$$ represent the number of bit strings of length n containing two consecutive 1s. What is the recurrence rela...
GATE CSE 2015 Set 1
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$$\sum\limits_{x = 1}^{99} {{1 \over {x\left( {x + 1} \right)}}} $$ = _____________.
GATE CSE 2015 Set 1
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If the following system has non - trivial solution $$$px+qy+rz=0$$$ $$$qx+ry+pz=0$$$ $$$rx+py+qz=0$$$ Then which one of...
GATE CSE 2015 Set 3
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Perform the following operations on the matrix $$\left[ {\matrix{ 3 & 4 & {45} \cr 7 & 9 & {105}...
GATE CSE 2015 Set 2
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Consider the following $$2 \times 2$$ matrix $$A$$ where two elements are unknown and are marked by $$a$$ and $$b.$$ The...
GATE CSE 2015 Set 1
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The product of the non-zero eigenvalues of the matrix $$\left[ {\matrix{ 1 & 0 & 0 & 0 & 1 \cr ...
GATE CSE 2014 Set 2
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$$\left[ A \right]$$ is a square matrix which is neither symmetric nor skew-symmetric and $${\left[ A \right]^T}$$ is it...
GATE CSE 2011
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Consider the matrix as given below. $$$\left[ {\matrix{ 1 & 2 & 3 \cr 0 & 4 & 7 \cr 0 &...
GATE CSE 2011
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Four matrices $${M_1},\,\,\,{M_2},\,\,\,{M_3}$$ and $${M_4}$$ of dimensions $$p\,\,x\,\,q,\,\,\,\,\,q\,\,x\,\,e,\,\,\,\,...
GATE CSE 2011
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Consider the following matrix $$A = \left[ {\matrix{ 2 & 3 \cr x & y \cr } } \right]\,\,$$ If the ei...
GATE CSE 2010
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If $$M$$ is a square matrix with a zero determinant, which of the following assertion(s) is (are) correct? $$S1$$ : Eac...
GATE CSE 2008
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How many of the following matrices have an eigen value $$1$$? $$\left[ {\matrix{ 1 & 0 \cr 0 & 0 \cr ...
GATE CSE 2008
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Let $$A$$ be $$a$$ $$4$$ $$x$$ $$4$$ matrix with eigen values $$-5$$, $$-2, 1, 4$$. Which of the following is an eigen ...
GATE CSE 2007
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What are the eigen values of the matrix $$P$$ given below? $$$P = \left( {\matrix{ a & 1 & 0 \cr 1 &...
GATE CSE 2006
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$$F$$ is an $$n$$ $$x$$ $$n$$ real matrix. $$b$$ is an $$n$$ $$x$$ $$1$$ real vector. Suppose there are two $$n$$ $$x$$ ...
GATE CSE 2006
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Consider the following system of equations in three real variables $$x1, x2$$ and $$x3$$ : $$2x1 - x2 + 3x3 = 1$$ $$3x1 ...
GATE CSE 2005
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What are the eigen values of the following $$2x2$$ matrix? $$$\left[ {\matrix{ 2 & { - 1} \cr { - 4} & 5...
GATE CSE 2005
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Consider the set $$H$$ of all $$3$$ $$X$$ $$3$$ matrices of the type $$$\left[ {\matrix{ a & f & e \cr 0...
GATE CSE 2005
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In an M$$ \times $$N matrix such that all non-zero entries are covered in $$a$$ rows and $$b$$ columns. Then the maximum...
GATE CSE 2004
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If matrix $$X = \left[ {\matrix{ a & 1 \cr { - {a^2} + a - 1} & {1 - a} \cr } } \right]$$ and $${X^...
GATE CSE 2004
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Let $$A$$ be and n$$ \times $$n matrix of the folowing form. What is the value of the determinant of $$A$$?...
GATE CSE 2004
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How many solutions does the following system of linear equations have? - x + 5y = - 1x - y = 2x + 3y = 3 ...
GATE CSE 2004
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Consider the following system of linear equations $$$\left[ {\matrix{ 2 & 1 & { - 4} \cr 4 & 3 &...
GATE CSE 2003
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Obtain the eigen values of the matrix $$$A = \left[ {\matrix{ 1 & 2 & {34} & {49} \cr 0 & 2 &...
GATE CSE 2002
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Consider the following determinant $$$\Delta = \left| {\matrix{ 1 & a & {bc} \cr 1 & a & {ca} ...
GATE CSE 1998
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The rank of the matrix given below is: $$$\left[ {\matrix{ 1 & 4 & 8 & 7 \cr 0 & 0 & 3 &...
GATE CSE 1998
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Let $$A = ({a_{ij}})$$ be and n-rowed square matrix and $${I_{12}}$$ be the matrix obtained by interchanging the first a...
GATE CSE 1997
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The matrices$$\left[ {\matrix{ {\cos \,\theta } & { - \sin \,\theta } \cr {\sin \,\,\theta } & {\cos \,\...
GATE CSE 1996
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In a compact single dimensional array representation for lower triangular matrices (i.e., all the elements above the dia...
GATE CSE 1994
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If A and B are real symmetric matrices of size n x n. Then, which one of the following is true?
GATE CSE 1994
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If a, b and c are constants, which of the following is a linear inequality?
GATE CSE 1987
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A square matrix is singular whenever:
GATE CSE 1987
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