GATE EE
Engineering Mathematics
Linear Algebra
Previous Years Questions

## Marks 1

Consider a 3 $$\times$$ 3 matrix A whose (i, j)-th element, ai,j = (i $$-$$ j)3. Then the matrix A will be
e4 denotes the exponential of a square matrix A. Suppose $$\lambda$$ is an eigen value and v is the corresponding eigen-vector of matrix A. Consider t...
Consider a matrix $$A = \left[ {\matrix{ 1 & 0 & 0 \cr 0 & 4 & { - 2} \cr 0 & 1 & 1 \cr } } \right]$$. The matrix A satisfies the equ...
The matrix $$A = \left[ {\matrix{ {{3 \over 2}} & 0 & {{1 \over 2}} \cr 0 & { - 1} & 0 \cr {{1 \over 2}} & 0 & {{...$$A3 \times 3$$matrix$$P$$is such that ,$${p^3} = P.$$Then the eigen values of$$P$$are Consider$$3 \times 3$$matrix with every element being equal to$$1.$$Its only non-zero eigenvalue is __________. We have a set of$$3$$linear equations in$$3$$unknown.$$'X \equiv Y'$$means$$X$$and$$Y$$are equivalent statements and$$'X \ne Y'$$means$$X...
If the sum of the diagonal elements of a $$2 \times 2$$ matrix is $$-6$$, then the maximum possible value of determinant of the matrix is ____________...
Which one of the following statements is true for all real symmetric matrices?
Given a system of equations $$x + 2y + 2z = {b_1}$$$$$5x + y + 3z = {b_2}$$$ Which of the following is true its solutions
Given that $$A = \left[ {\matrix{ { - 5} & { - 3} \cr 2 & 0 \cr } } \right]$$ and $${\rm I} = \left[ {\matrix{ 1 & 0 \cr ... An eigen vector of$$p = \left[ {\matrix{ 1 & 1 & 0 \cr 0 & 2 & 2 \cr 0 & 0 & 3 \cr } } \right]$$is The trace and determinant of a$$2 \times 2$$matrix are shown to be$$-2$$and$$-35$$respectively. Its eigen values are The characteristic equation of a$$3\,\, \times \,\,3$$matrix$$P$$is defined as$$\alpha \left( \lambda \right) = \left| {\lambda {\rm I} - P} \r...
$$A$$ is $$m$$ $$x$$ $$n$$ full rank matrix with $$m > n$$ and $${\rm I}$$ is an identity matrix. Let matrix $${A^ + } = {\left( {{A^T}A} \right)^{...$$X = {\left[ {\matrix{ {{x_1}} & {{x_2}} & {.......\,{x_n}} \cr } } \right]^T}$$is an$$n$$-tuple non- zero vector. The$$n\,\, \time...
In the matrix equation $$PX=Q$$ which of the following is a necessary condition for the existence of atleast one solution for the unknown vector $$X.... The determinant of the matrix$$\left[ {\matrix{ 1 & 0 & 0 & 0 \cr {100} & 1 & 0 & 0 \cr {100} & {200} & ...
If $$A = \left[ {\matrix{ 1 & { - 2} & { - 1} \cr 2 & 3 & 1 \cr 0 & 5 & { - 2} \cr } } \right]$$ and $$adj (A... Find the eigen values and eigen vectors of the matrix$$\left[ {\matrix{ 3 & { - 1} \cr { - 1} & 3 \cr } } \right]A = \left[ {\matrix{ 2 & 0 & 0 & { - 1} \cr 0 & 1 & 0 & 0 \cr 0 & 0 & 3 & 0 \cr { - 1} & 0...
A set of linear equations is represented by the matrix equations $$Ax=b.$$ The necessary condition for the existence of a solution for this system is
If the vector $$\left[ {\matrix{ 1 \cr 2 \cr { - 1} \cr } } \right]$$ is an eigen vector of $$A = \left[ {\matrix{ { - 2} & 2 ... If$$A = \left[ {\matrix{ 5 & 0 & 2 \cr 0 & 3 & 0 \cr 2 & 0 & 1 \cr } } \right]$$then$${A^{ - 1}} = $$Express the given matrix$$A = \left[ {\matrix{ 2 & 1 & 5 \cr 4 & 8 & {13} \cr 6 & {27} & {31} \cr } } \right...
The inverse of the matrix $$S = \left[ {\matrix{ 1 & { - 1} & 0 \cr 1 & 1 & 1 \cr 0 & 0 & 1 \cr } } \right]$$...
Given the matrix $$A = \left[ {\matrix{ 0 & 1 & 0 \cr 0 & 0 & 1 \cr { - 6} & { - 11} & { - 6} \cr } } \right]... The rank of the following$$(n+1)x(n+1)$$matrix, where$$'a'$$is a real number is$$$\left[ {\matrix{ 1 & a & {{a^2}} & . ... The eigen values of the matrix $$\left[ {\matrix{ a & 1 \cr a & 1 \cr } } \right]$$ are $$A$$ $$\,\,5 \times 7$$ matrix has all its entries equal to $$1.$$ Then the rank of a matrix is The number of linearly independent solutions of the system of equations $$\left[ {\matrix{ 1 & 0 & 2 \cr 1 & { - 1} & 0 \cr ... ## Marks 2 The eigen values of the matrix given below are$$\left[ {\matrix{ 0 & 1 & 0 \cr 0 & 0 & 1 \cr 0 & { - 3} & { - 4}... Let $$P = \left[ {\matrix{ 3 & 1 \cr 1 & 3 \cr } } \right].$$ Consider the set $$S$$ of all vectors $$\left( {\matrix{ x \cr ... Let$$A$$be a$$4 \times 3$$real matrix which rank$$2.$$Which one of the following statement is TRUE? Let the eigenvalues of a$$2 \times 2$$matrix$$A$$be$$1,-2$$with eigenvectors$${x_1}$$and$${x_2}$$respectively. Then the eigenvalues and eige... The maximum value of$$'a'$$such that the matrix$$\left[ {\matrix{ { - 3} & 0 & { - 2} \cr 1 & { - 1} & 0 \cr 0 & a... $$A = \left[ {\matrix{ p & q \cr r & s \cr } } \right];B = \left[ {\matrix{ {{p^2} + {q^2}} & {pr + qs} \cr {pr + qs} ... A system matrix is given as follows$$$A = \left[ {\matrix{ 0 & 1 & { - 1} \cr { - 6} & { - 11} & 6 \cr { - 6} & { -...
The equation $$\left[ {\matrix{ 2 & { - 2} \cr 1 & { - 1} \cr } } \right]\left[ {\matrix{ {{x_1}} \cr {{x_2}} \cr } } ... A matrix has eigen values$$-1$$and$$-2.$$The corresponding eigenvectors are$$\left[ {\matrix{ 1 \cr { - 1} \cr } } \right]$$and$$\l...
The two vectors $$\left[ {\matrix{ 1 & 1 & 1 \cr } } \right]$$ and $$\left[ {\matrix{ 1 & a & {{a^2}} \cr } } \right]$$ ...
The matrix $$\left[ A \right] = \left[ {\matrix{ 2 & 1 \cr 4 & { - 1} \cr } } \right]$$ is decomposed into a product of lower tria...
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 following statement is true ...
If the rank of a $$5x6$$ matrix $$Q$$ is $$4$$ then which one of the following statements is correct?
Let $$P$$ be $$2x2$$ real orthogonal matrix and $$\overline x$$ is a real vector $${\left[ {\matrix{ {{x_1}} & {{x_2}} \cr } } \right]^T}...$${q_1},\,{q_2},{q_3},.......{q_m}$$are$$n$$-dimensional vectors with$$m < n.$$This set of vectors is linearly dependent.$$Q$$is the matrix w... If$$A = \left[ {\matrix{ { - 3} & 2 \cr { - 1} & 0 \cr } } \right]\,$$then$${A^9}$$equals If$$A = \left[ {\matrix{ { - 3} & 2 \cr { - 1} & 0 \cr } } \right]$$then$$A$$satisfies the relation Let$$x$$and$$y$$be two vectors in a$$3-$$dimensional space and$$ < x,y > $$denote their dot product. Then the determinant det$$\left[ {...
For the matrix $$P = \left[ {\matrix{ 3 & { - 2} & 2 \cr 0 & { - 2} & 1 \cr 0 & 0 & 1 \cr } } \right],$$ one ...
If $$R = \left[ {\matrix{ 1 & 0 & { - 1} \cr 2 & 1 & { - 1} \cr 2 & 3 & 2 \cr } } \right]$$ then the top row...
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