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Test: Eigenvalues & Eigenvectors - 1 - Engineering Mathematics MCQ


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20 Questions MCQ Test Engineering Mathematics - Test: Eigenvalues & Eigenvectors - 1

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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

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 7

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 8

The number of linearly independent eigenvectors of 

Test: Eigenvalues & Eigenvectors - 1 - Question 9

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

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

 

Test: Eigenvalues & Eigenvectors - 1 - Question 10

One of the Eigenvectors of the matrix A =  is

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

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

So we can check by multiplication.

  

Test: Eigenvalues & Eigenvectors - 1 - Question 11

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 11

Test: Eigenvalues & Eigenvectors - 1 - Question 12

 

The largest eigenvalue of the matrix 
 is 

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

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 13

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 13

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

Test: Eigenvalues & Eigenvectors - 1 - Question 14

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 14

Test: Eigenvalues & Eigenvectors - 1 - Question 15

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

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

Test: Eigenvalues & Eigenvectors - 1 - Question 16

Let A be an n x 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 16

Answer: b) of modulus one

Explanation:

Since A is self-adjoint, its eigenvalues (denoted λ) are real.

Let v be an eigenvector with A v = λ v. Then
(A minus i I) times the inverse of (A plus i I), acting on v, gives
((λ minus i) divided by (λ plus i)) times v.

Therefore, every eigenvalue of (A − i I)(A + i I)⁻¹ has modulus one.

  • Observe that the absolute value of (λ minus i) is √(λ² + 1), which is the same as the absolute value of (λ plus i). Hence the absolute value of (λ minus i)/(λ plus i) is 1.

Test: Eigenvalues & Eigenvectors - 1 - Question 17

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 17

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 18

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 18

Test: Eigenvalues & Eigenvectors - 1 - Question 19

Test: Eigenvalues & Eigenvectors - 1 - Question 20

In matrix equation [A]{X} = {R}, one of the eigenvalues of matrix A is

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

Vector R may be written as

R = 32i + 16j + 64k = 16 (2i + j + 4k) = 16X

So vector X after transformation remains on its span and only its magnitude is enlarged by 16. Hence, one of eigenvalue of matrix A is 16.

Hence option D is the correct option.

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