GATE CSE
Discrete Mathematics
Linear Algebra
Previous Years Questions

Marks 1

Let $$A = \left[ {\matrix{ 1 & 2 & 3 & 4 \cr 4 & 1 & 2 & 3 \cr 3 & 4 & 1 & 2 \cr 2 & 3 & 4 & 1 \cr } } \right]$$ and $$B = \left... Consider the following two statements with respect to the matrices Am$$\times$$n , Bn$$\times$$m , Cn$$\times$$n and Dn$$\times$$n . Statement... Which one of the following is the closed form for the generating function of the sequence (an}n$$\ge$$0 defined below?$${a_n} = \left\{ {\matrix{ ...
Consider solving the following system of simultaneous equations using LU decomposition. x1 + x2 $$-$$ 2x3 = 4 x1 + 3x2 $$-$$ x3 = 7 2x1 + x2 $$-$$ 5x3...
Which of the following is/are the eigen vector(s) for the matrix given below? $$\left( {\matrix{ { - 9} & { - 6} & { - 2} & { - 4} \cr { - 8} ... Let X be a square matrix. Consider the following two statements on X. I. X is invertible. II. Determinant of X is non-zero. Which one of the followin... Consider a matrix$$A = u{v^T}$$where$$u = \left( {\matrix{ 1 \cr 2 \cr } } \right),v = \left( {\matrix{ 1 \cr 1 \cr } } \ri...
Let $${c_1},.....,\,\,{c_n}$$ be scalars, not all zero, such that $$\sum\limits_{i = 1}^n {{c_i}{a_i} = 0}$$ where $${{a_i}}$$ are column vectors in ...
Let $$P = \left[ {\matrix{ 1 & 1 & { - 1} \cr 2 & { - 3} & 4 \cr 3 & { - 2} & 3 \cr } } \right]$$ and $$Q = \... Suppose that the eigen values of matrix$$A$$are$$1, 2, 4.$$The determinant of$${\left( {{A^{ - 1}}} \right)^T}$$is _______. Consider the system, each consisting of m linear equations in$$n$$variables.$$I.\,\,\,$$If$$m < n,$$then all such system have a solutio... Two eigenvalues of a$$3 \times 3$$real matrix$$P$$are$$\left( {2 + \sqrt { - 1} } \right)$$and$$3.$$The determinant of$$P$$is _______. Let$${a_n}$$be the number of$$n$$-bit strings that do NOT contain two consecutive$$1s.$$Which one of the following is the recurrence relation for... In the LU decomposition of the matrix$$\left[ {\matrix{ 2 & 2 \cr 4 & 9 \cr } } \right]$$, if the diagonal elements of U are both... The larger of the two eigenvalues of the matrix$$\left[ {\matrix{ 4 & 5 \cr 2 & 1 \cr } } \right]$$is ______. The number of divisors of$$2100$$is ___________. In the given matrix$$\left[ {\matrix{ 1 & { - 1} & 2 \cr 0 & 1 & 0 \cr 1 & 2 & 1 \cr } } \right],$$one of t... The value of the dot product of the eigenvectors corresponding to any pair of different eigen values of a 4-by-4 symmetric positive definite matrix is... Consider the following system of equations: 3x + 2y = 1 4x + 7z = 1 x + y + z =3 x - 2y + 7z = 0 The number of solutions for this system is _____... Which one of the following statements is TRUE about every$$n\,\, \times \,n$$matrix with only real eigen values? If$${V_1}$$and$${V_2}$$are 4-dimensional subspaces of a 6-dimensional vector space V, then the smallest possible dimension of$${V_1}\, \cap \,\,{...
If the matrix A is such that $$A = \left[ {\matrix{ 2 \cr { - 4} \cr 7 \cr } } \right]\,\,\left[ {\matrix{ 1 & 9 & 5 \c... Which of the following does not equal$$\left| {\matrix{ 1 & x & {{x^2}} \cr 1 & y & {{y^2}} \cr 1 & z & {{z^2}}...
Let $$A$$ be the $$2 \times 2$$ matrix with elements $${a_{11}} = {a_{12}} = {a_{21}} = + 1$$ and $${a_{22}} = - 1$$. Then the eigen values of the m...
Consider the following matrix $$A = \left[ {\matrix{ 2 & 3 \cr x & y \cr } } \right].$$ If the eigen values of $$A$$ are $$4$$ an...
The following system of equations $${x_1}\, + \,{x_2}\, + 2{x_3}\, = 1$$ $${x_1}\, + \,2 {x_2}\, + 3{x_3}\, = 2$$ $${x_1}\, + \,4{x_2}\, + a{x_3}\,... Let$$A$$be the matrix$$\left[ {\matrix{ 3 & 1 \cr 1 & 2 \cr } } \right]$$. What is the maximum value of$${x^T}Ax$$where the ... The determination of the matrix given below is$$$\left[ {\matrix{ 0 & 1 & 0 & 2 \cr { - 1} & 1 & 1 & 3 \cr 0 &am... Let A, B, C, D be $$n\,\, \times \,\,n$$ matrices, each with non-zero determination. If ABCD = I, then $${B^{ - 1}}$$ is What values of x, y and z satisfy the following system of linear equations? $$\left[ {\matrix{ 1 & 2 & 3 \cr 1 & 3 & 4 \cr ... The number of different$$n \times n$$symmetric matrices with each elements being either$$0$$or$$1$$is$$A$$system of equations represented by$$AX=0$$where$$X$$is a column vector of unknown and$$A$$is a square matrix containing coefficients has a... The rank of the matrix$$\left[ {\matrix{ 1 & 1 \cr 0 & 0 \cr } } \right]\,\,is$$Consider the following statements: S1: The sum of two singular n x n matrices may be non-singular S2: The sum of two n x n non-singular matrices may... The determinant of the matrix$$$\left[ {\matrix{ 2 & 0 & 0 & 0 \cr 8 & 1 & 7 & 2 \cr 2 & 0 & 2 & 0 ...
An $$n\,\, \times \,\,n$$ array v is defined as follows v[i, j] = i - j for all i, j, $$1\,\, \le \,\,i\,\, \le \,\,n,\,1\,\, \le \,\,j\,\, \le \,\,n... Consider the following set a equations x + 2y = 5 4x + 8y = 12 3x + 6y + 3z = 15 This set The determination of the matrix$$$\left[ {\matrix{ 6 & { - 8} & 1 & 1 \cr 0 & 2 & 4 & 6 \cr 0 & 0 & 4 &a... 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 and X is a n x 1 column vector o... Let $$A = \left[ {\matrix{ {{a_{11}}} & {{a_{12}}} \cr {{a_{21}}} & {{a_{22}}} \cr } } \right]\,\,$$ and $$B = \left[ {\matrix{ ... The rank of the following (n + 1) x (n + 1) matrix, where a is a real number is$$$\left[ {\matrix{ 1 & a & {{a^2}} & . & . & ...
The rank of the following (n + 1) x (n + 1) matrix, where a is a real number is $$\left[ {\matrix{ 1 & a & {{a^2}} & . & . & ... The inverse of the matrix$$\left[ {\matrix{ 1 & 0 & 1 \cr { - 1} & 1 & 1 \cr 0 & 1 & 0 \cr } } \right]$$is The rank of the matrix$$\left[ {\matrix{ 0 & 0 & { - 3} \cr 9 & 3 & 5 \cr 3 & 1 & 1 \cr } } \right]$$is If$$A = \left[ {\matrix{ 1 & 0 & 0 & 1 \cr 0 & { - 1} & 0 & { - 1} \cr 0 & 0 & i & i \cr 0 &...
The eigen vector (s) of the matrix $$\left[ {\matrix{ 0 & 0 & \alpha \cr 0 & 0 & 0 \cr 0 & 0 & 0 \cr } } \r... Marks 2 Consider the following matrix.$$\left( {\begin{array}{*{20}{c}} 0&1&1&1\\ 1&0&1&1\\ 1&1&0&1\\ 1&1&1&...
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 a matrix M, respectively. Cons...
Which one of the following is a closed form expression for the generating function of the sequence $$\left\{ {{a_n}} \right\},$$ where $${a_n} = 2n + ... Consider a matrix P whose only eigenvectors are the multiples of$$\left[ {\matrix{ 1 \cr 4 \cr } } \right].$$Consider the following sta... Let$$A$$be$$n\,\, \times \,\,n$$real valued square symmetric matrix of rank$$2$$with$$\sum\limits_{i = 1}^n {\sum\limits_{j = 1}^n {A_{ij}^2 = ...
If the characteristic polynomial of a $$3 \times 3$$ matrix $$M$$ over $$R$$(the set of real numbers) is $${\lambda ^3} - 4{\lambda ^2} + a\lambda + ... Let$${A_1},\,{A_2},\,{A_3}$$and$${A_4}$$be four matrices of dimensions$$10 \times 5,\,5 \times 20,\,20 \times 10,$$and$$10 \times 5,$$respecti... The value of the expression$${13^{99}}$$($$mod17$$), in the range$$0$$to$$16,$$is ______________ . Consider the recurrence relation$${a_1} = 8,\,{a_n} = 6{n^2} + 2n + {a_{n - 1}}.$$Let$${a_{99}} = K \times {10^4}.$$The value of$$K$$is _______... Consider the following$$2 \times 2$$matrix$$A$$where two elements are unknown and are marked by$$a$$and$$b.$$The eigenvalues of this matrix a...$$\sum\limits_{x = 1}^{99} {{1 \over {x\left( {x + 1} \right)}}} $$= _____________. Let$${a_n}$$represent the number of bit strings of length n containing two consecutive 1s. What is the recurrence relation for$${a_n}$$? Perform the following operations on the matrix$$\left[ {\matrix{ 3 & 4 & {45} \cr 7 & 9 & {105} \cr {13} & 2 & {...
If the following system has non - trivial solution $$px+qy+rz=0$$$$$qx+ry+pz=0$$$ $$rx+py+qz=0$$$Then which one of the following Options is TRUE... The product of the non-zero eigenvalues of the matrix $$\left[ {\matrix{ 1 & 0 & 0 & 0 & 1 \cr 0 & 1 & 1 & 1 &... Four matrices$${M_1},\,\,\,{M_2},\,\,\,{M_3}$$and$${M_4}$$of dimensions$$p\,\,x\,\,q,\,\,\,\,\,q\,\,x\,\,e,\,\,\,\,\,r\,\,x\,\,s$$and$$\,\,\,\,... Consider the matrix as given below. $$\left[ {\matrix{ 1 & 2 & 3 \cr 0 & 4 & 7 \cr 0 & 0 & 3 \cr } } \right...$$\left[ A \right]$$is a square matrix which is neither symmetric nor skew-symmetric and$${\left[ A \right]^T}$$is its transpose. The sum and diffe... Consider the following matrix$$A = \left[ {\matrix{ 2 & 3 \cr x & y \cr } } \right]\,\,$$If the eigen values of$$A$$are$$4$$... How many of the following matrices have an eigen value$$1$$?$$\left[ {\matrix{ 1 & 0 \cr 0 & 0 \cr } } \right],\,\,\left[ {\matr... If $$M$$ is a square matrix with a zero determinant, which of the following assertion(s) is (are) correct? $$S1$$ : Each row of $$M$$ can be represen... Let $$A$$ be $$a$$ $$4$$ $$x$$ $$4$$ matrix with eigen values $$-5$$, $$-2, 1, 4$$. Which of the following is an eigen value of $$\left[ {\matrix{ ...$$F$$is an$$nxn$$real matrix.$$b$$is an$$nx1$$real vector. Suppose there are two$$nx1$$vectors,$$u$$and$$v$$... What are the eigen values of the matrix$$P$$given below?$$$P = \left( {\matrix{ a & 1 & 0 \cr 1 & a & 1 \cr 0 & 1 ...
What are the eigen values of the following $$2x2$$ matrix? $$\left[ {\matrix{ 2 & { - 1} \cr { - 4} & 5 \cr } } \right]$$$Consider the set $$H$$ of all $$3$$ $$X$$ $$3$$ matrices of the type $$\left[ {\matrix{ a & f & e \cr 0 & b & d \cr 0 &a... Consider the following system of equations in three real variables$$x1, x2$$and$$x3$$:$$2x1 - x2 + 3x3 = 13x1 + 2x2 + 5x3 = 2 - x1 + 4x... How many solutions does the following system of linear equations have? - x + 5y = - 1x - y = 2x + 3y = 3 In an M$$\times$$N matrix such that all non-zero entries are covered in $$a$$ rows and $$b$$ columns. Then the maximum number of non-zero entries, s... If matrix $$X = \left[ {\matrix{ a & 1 \cr { - {a^2} + a - 1} & {1 - a} \cr } } \right]$$ and $${X^2} - X + 1 = 0$$ ($${\rm I}$$ ... Let $$A$$ be and n$$\times$$n matrix of the folowing form. What is the value of the determinant of $$A$$?... Consider the following system of linear equations $$\left[ {\matrix{ 2 & 1 & { - 4} \cr 4 & 3 & { - 12} \cr 1 & 2 &... Obtain the eigen values of the matrix$$$A = \left[ {\matrix{ 1 & 2 & {34} & {49} \cr 0 & 2 & {43} & {94} \cr 0 &...
The rank of the matrix given below is: $$\left[ {\matrix{ 1 & 4 & 8 & 7 \cr 0 & 0 & 3 & 0 \cr 4 & 2 & 3 ... Consider the following determinant$$$\Delta = \left| {\matrix{ 1 & a & {bc} \cr 1 & a & {ca} \cr 1 & a & {ab} ... Let $$A = ({a_{ij}})$$ be and n-rowed square matrix and $${I_{12}}$$ be the matrix obtained by interchanging the first and second rows of the n-rowed ... The matrices$$\left[ {\matrix{ {\cos \,\theta } & { - \sin \,\theta } \cr {\sin \,\,\theta } & {\cos \,\,\theta } \cr } } \right]\... If A and B are real symmetric matrices of size n x n. Then, which one of the following is true? In a compact single dimensional array representation for lower triangular matrices (i.e., all the elements above the diagonal are zero) of size$$n$\$ ...
A square matrix is singular whenever:
If a, b and c are constants, which of the following is a linear inequality?
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