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Test: Eigenvalues & Eigenvectors - 2


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25 Questions MCQ Test GATE Mechanical (ME) 2023 Mock Test Series | Test: Eigenvalues & Eigenvectors - 2

Test: Eigenvalues & Eigenvectors - 2 for Civil Engineering (CE) 2022 is part of GATE Mechanical (ME) 2023 Mock Test Series preparation. The Test: Eigenvalues & Eigenvectors - 2 questions and answers have been prepared according to the Civil Engineering (CE) exam syllabus.The Test: Eigenvalues & Eigenvectors - 2 MCQs are made for Civil Engineering (CE) 2022 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Test: Eigenvalues & Eigenvectors - 2 below.
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Test: Eigenvalues & Eigenvectors - 2 - Question 1

For the matrix  the eigen value corresponding to the eigenvector 

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*Multiple options can be correct
Test: Eigenvalues & Eigenvectors - 2 - Question 2

The eigen values of the matrix  

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Test: Eigenvalues & Eigenvectors - 2 - Question 3

For the matrix    the eigen value are

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Test: Eigenvalues & Eigenvectors - 2 - Question 4

For which value of x will the matrix given below become singular? 

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Let the given matrix be A.  A is singular. 

Test: Eigenvalues & Eigenvectors - 2 - Question 5

If a square matrix A is real and symmetric, then the eigenvaluesn 

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Test: Eigenvalues & Eigenvectors - 2 - Question 6

The matrix    has one eigenvalue equal to 3. The sum of the other two eigenvalues is 

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Let the given matrix be  A.  

Test: Eigenvalues & Eigenvectors - 2 - Question 7

For a matrix    the transpose of the matrix is equal to the inverse of the  matrix,     The value of x is given by 

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Test: Eigenvalues & Eigenvectors - 2 - Question 8

The eigen values of the matrix  

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Test: Eigenvalues & Eigenvectors - 2 - Question 9

For a given matrix      one of the eigenvalues is 3. The other two eigenvalues are 

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Test: Eigenvalues & Eigenvectors - 2 - Question 10

The Eigen values of the matrix 

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Test: Eigenvalues & Eigenvectors - 2 - Question 11

In the matrix equation Px = q which of the following is a necessary condition for the existence of at least one solution for the unknown vector x:  

Test: Eigenvalues & Eigenvectors - 2 - Question 12

If    then top row of R-1 is 

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Test: Eigenvalues & Eigenvectors - 2 - Question 13

Cayley - Hamiltion Theorem states that square matrix satisfies its own characteristic equation, Consider a matrix 

A satisfies the relation 

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Characteristic equation of A is  

Test: Eigenvalues & Eigenvectors - 2 - Question 14

The characteristic equation of a (3×3) matrix P is defined as       If I denote identity matrix, then the inverse of matrix P will be 

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Given ch. equof A is 

Test: Eigenvalues & Eigenvectors - 2 - Question 15

Let P be a 2×2 real orthogonal matrix and  s a real vector    with length   Then which one of the following statements is correct?  

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Test: Eigenvalues & Eigenvectors - 2 - Question 16

An eigenvector of 

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Eigen values of P are 1,2,3 

Test: Eigenvalues & Eigenvectors - 2 - Question 17

Let A be an n × n real matrix such that A2 = I and y = be an n – dimensional vector.  Then the linear system of equations Ax = y has 

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By Cramer’s rule AX =y has unique solution. 

Test: Eigenvalues & Eigenvectors - 2 - Question 18

A real n × n matrix A = {aij} is defined as follows: 

aij = i = 0, if 

  i  = j, otherwise 

The summation of all n eigen values of A is  

Detailed Solution for Test: Eigenvalues & Eigenvectors - 2 - Question 18

 It’s a diagonal marix diagonal contain’s  n elements 1,2,----,n. 

As diagonal elements are eigen valves.

Test: Eigenvalues & Eigenvectors - 2 - Question 19

The following system of equations  

has a unique solution. The only possible value(s) for a is/are 

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 System has unique Soln if  rank (A) = rank ( A ) = 3 . It is possible if a ≠ 5.

Test: Eigenvalues & Eigenvectors - 2 - Question 20

The eigenvalues of

are

Detailed Solution for Test: Eigenvalues & Eigenvectors - 2 - Question 20

The eigenvalues of an upper triangular matrix are simply the diagonal entries of the matrix. Hence 5, -19, and 37 are the eigenvalues of the matrix. Alternately, look at 

Then  = 5,-19,37 are the roots of the equation; and hence, the eigenvalues of [A].

Test: Eigenvalues & Eigenvectors - 2 - Question 21

The number of different n × n symmetric matrices with each element being either 0 or 1 is: (Note : power (2, x) is same as 2x) 

Detailed Solution for Test: Eigenvalues & Eigenvectors - 2 - Question 21

In a symmetric matrix, the lower triangle must be the minor image of upper triangle using the diagonal as mirror. Diagonal elements may be anything. Therefore, when we are counting symmetric matrices we count how many ways are there to fill the upper triangle and diagonal elements. Since the first row has n elements, second (n – 1) elements, third row (n – 2) elements and so on upto last row, one element.  Total number of elements in diagonal + upper triangle  

Now, each one of these elements can be either 0 or 1. So that number of ways we can fill these elements is

Since there is no choice for lower triangle elements the answer is power   which 
is choice (c). 

Test: Eigenvalues & Eigenvectors - 2 - Question 22

In an M × N matrix such that all non-zero entries are covered in a rows and b column. Then the maximum number of non-zero entries, such that no two are on the same row or column, is 

Detailed Solution for Test: Eigenvalues & Eigenvectors - 2 - Question 22

Suppose a < b, for example let a = 3, b= 5, then we can put non-zero entries only in 3 rows and 5 columns. So suppose we put non-zero entries in any 3 rows in 3 different columns. Now we can’t put any other non-zero entry anywhere in matrix, because if we put it in some other row, then we will have 4 rows containing non-zeros, if we put it in one of those 3 rows, then we will have more than one non-zero entry in one row, which is not allowed.

So we can fill only “a” non-zero entries if a < b, similarly if b < a, we can put only “b” non-zero entries. So answer is ≤min(a,b), because whatever is less between a and b, we can put atmost that many non-zero entries.

Test: Eigenvalues & Eigenvectors - 2 - Question 23

Consider the following system of equation in three real variables x1, x2 and x3

This system of equations has 

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∴ Rank (A)= Rank ( A ) = 3

Test: Eigenvalues & Eigenvectors - 2 - Question 24

How many of the following matrics have an eigenvalue 1?  

 

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Rest given matrix are triangular matrix. so diagonal elements are the eigen  values. 

Test: Eigenvalues & Eigenvectors - 2 - Question 25

What are the eigen values of the following 2 × 2 matrix?  

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