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Test: Eigenvalues & Eigenvectors - 1 - Mechanical Engineering MCQ


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20 Questions MCQ Test GATE Mechanical (ME) 2024 Mock Test Series - Test: Eigenvalues & Eigenvectors - 1

Test: Eigenvalues & Eigenvectors - 1 for Mechanical Engineering 2024 is part of GATE Mechanical (ME) 2024 Mock Test Series preparation. The Test: Eigenvalues & Eigenvectors - 1 questions and answers have been prepared according to the Mechanical Engineering exam syllabus.The Test: Eigenvalues & Eigenvectors - 1 MCQs are made for Mechanical Engineering 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Test: Eigenvalues & Eigenvectors - 1 below.
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Test: Eigenvalues & Eigenvectors - 1 - Question 1

Find the sum of the Eigenvalues of the matrix

Detailed Solution for Test: Eigenvalues & Eigenvectors - 1 - Question 1

According to the property of the Eigenvalues, the sum of the Eigenvalues of a matrix is its trace that is the sum of the elements of the principal diagonal. 
Therefore, the sum of the Eigenvalues = 3 + 4 + 1 = 8.

Test: Eigenvalues & Eigenvectors - 1 - Question 2

Find the Eigenvalues of matrix

Detailed Solution for Test: Eigenvalues & Eigenvectors - 1 - Question 2

A - λI = 0

Now, After taking the determinant:

(4 - λ)2 - 1 = 0

16 + λ2 - 8λ - 1 = 0

λ2 - 8λ + 15 = 0

 - 3) (λ - 5) = 0

λ = 3, 5

Test: Eigenvalues & Eigenvectors - 1 - Question 3

All the four entries of the 2 × 2 matrix    are nonzero, and one of its eigen values is zero. Which of the following statements is true?

Detailed Solution for Test: Eigenvalues & Eigenvectors - 1 - Question 3

One eigen value is zero

Test: Eigenvalues & Eigenvectors - 1 - Question 4

The eigen values of the following matrix are 

Detailed Solution for Test: Eigenvalues & Eigenvectors - 1 - Question 4

Let the matrix be A.  We know, Trace (A) = sum of eigen values. 

Test: Eigenvalues & Eigenvectors - 1 - Question 5

The three characteristic roots of the following matrix A  

Detailed Solution for Test: Eigenvalues & Eigenvectors - 1 - Question 5

Test: Eigenvalues & Eigenvectors - 1 - Question 6

The sum of the eigenvalues of the matrix given below is

Detailed Solution for Test: Eigenvalues & Eigenvectors - 1 - Question 6

Test: Eigenvalues & Eigenvectors - 1 - Question 7

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

Detailed Solution for Test: Eigenvalues & Eigenvectors - 1 - Question 7

Let the given matrix be A.  A is singular. 

Test: Eigenvalues & Eigenvectors - 1 - Question 8

Eigen values of a matrix    are 5 and 1. What are the eigen values of the matrix S2  = SS?

Detailed Solution for Test: Eigenvalues & Eigenvectors - 1 - Question 8

We know If λ be the eigen value of A ⇒λ2 is an eigen value of A2 .

For S matrix, if eigen values are λ1, λ2, λ3,... then for S² matrix, the eigen values will be λ²1 λ²2 λ²3......
For S matrix, if eigen values are 1 and 5 then for S² matrix, the eigen values are 1 and 25

Test: Eigenvalues & Eigenvectors - 1 - Question 9

The number of linearly independent eigenvectors of 

Detailed Solution for Test: Eigenvalues & Eigenvectors - 1 - Question 9

Number of linear independent vectors is equal to the sum of Geometric Multiplicity of eigen values. Here only eigen value is 2.

To find Geometric multiplicity find n-r of (matrix-2I), where n is order and r is rank.

Rank of obtained matrix is 1 and n = 2 so n-r = 1. Therefore the no of linearly independent eigen vectors is 1

Test: Eigenvalues & Eigenvectors - 1 - Question 10

The eigenvectors of the matrix     are written in the form  . What is a + b? 

Detailed Solution for Test: Eigenvalues & Eigenvectors - 1 - Question 10

 

Test: Eigenvalues & Eigenvectors - 1 - Question 11

One of the Eigenvectors of the matrix A =  is

Detailed Solution for Test: Eigenvalues & Eigenvectors - 1 - Question 11

The eigen vectors of A are given by  AX= λ X  

So we can check by multiplication.

  

Test: Eigenvalues & Eigenvectors - 1 - Question 12

The minimum and the maximum eigen values of the matrix    are –2 and 6, respectively. What  is the other eigen value?  

Detailed Solution for Test: Eigenvalues & Eigenvectors - 1 - Question 12

Test: Eigenvalues & Eigenvectors - 1 - Question 13

 

The largest eigenvalue of the matrix 
 is 

Detailed Solution for Test: Eigenvalues & Eigenvectors - 1 - Question 13

Concept:

If A is any square matrix of order n, we can form the matrix [A – λI], where I is the nth order unit matrix.

The determinant of this matrix equated to zero

i.e. |A – λI| = 0 is called the characteristic equation of A.

The roots of the characteristic equation are called Eigenvalues or latent roots or characteristic roots of matrix A.

Test: Eigenvalues & Eigenvectors - 1 - Question 14

The state variable description of a linear autonomous system is, X= AX, 

Where X is the two dimensional state vector and A is the system matrix given by 

The roots of the characteristic equation are 

Detailed Solution for Test: Eigenvalues & Eigenvectors - 1 - Question 14

Characteristic equation will be :λ2 -4 =0 thus root of characteristic equation will be +2 and - 2.

Test: Eigenvalues & Eigenvectors - 1 - Question 15

For the matrix   s one of the eigen values is equal to -2. Which of the following  is an eigen vector? 

Detailed Solution for Test: Eigenvalues & Eigenvectors - 1 - Question 15

Test: Eigenvalues & Eigenvectors - 1 - Question 16

x=[x1x2…..xn]T is an n-tuple nonzero vector. The n×n matrix V=xxT    

Detailed Solution for Test: Eigenvalues & Eigenvectors - 1 - Question 16

Test: Eigenvalues & Eigenvectors - 1 - Question 17

Let A be an n × n complex matrix. Assume that A is self-adjoint and let B denotes the inverse of (A + iIn). Then all eigenvalues of (A - iIn)B are 

Detailed Solution for Test: Eigenvalues & Eigenvectors - 1 - Question 17

Concept: 

(i) A complex matrix A is said to be self-adjoint if A = A* where A* = A-T

(ii) Eigenvalues of a self-adjoint matrix are real

Explanation:

A is an n × n complex matrix and A is self-adjoint so

A = A*

B denotes the inverse of (A + iIn).

So B = (A + iIn)-1

Let λ be an eigenvalue of A

then λ + i is an eigenvalue of A + iIn

and (λ + i)-1 is an eigenvalue of B

Also λ - i is an eigenvalue of A - iIn

Hence eigenvalue of (A - iIn)B

= (λ - i)(λ + i)-1 

Test: Eigenvalues & Eigenvectors - 1 - Question 18

If the rank of a (5×6) matrix Q is 4, then which one of the following statements is correct?  

Detailed Solution for Test: Eigenvalues & Eigenvectors - 1 - Question 18

Rank of a matrix is equal to the No. of linearly independent row or no. of linearly  independent column vector. 

Test: Eigenvalues & Eigenvectors - 1 - Question 19

The trace and determinate of a 2 ×2 matrix are known to be – 2 and – 35 respectively. Its eigenvalues are 

Detailed Solution for Test: Eigenvalues & Eigenvectors - 1 - Question 19

Test: Eigenvalues & Eigenvectors - 1 - Question 20

Identify which one of the following is an eigenvector of the matrix  

Detailed Solution for Test: Eigenvalues & Eigenvectors - 1 - Question 20

Eigen Value (λ ) are 1,− 2.

 be the eigen  of A. Corresponding 

To λ then. 

be the eigen vector corrosponding to λ = 1

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