Test: Engineering Mathematics- 3


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10 Questions MCQ Test RRB JE for Computer Science Engineering | Test: Engineering Mathematics- 3

Test: Engineering Mathematics- 3 for Computer Science Engineering (CSE) 2022 is part of RRB JE for Computer Science Engineering preparation. The Test: Engineering Mathematics- 3 questions and answers have been prepared according to the Computer Science Engineering (CSE) exam syllabus.The Test: Engineering Mathematics- 3 MCQs are made for Computer Science Engineering (CSE) 2022 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Test: Engineering Mathematics- 3 below.
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Test: Engineering Mathematics- 3 - Question 1

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

Detailed Solution for Test: Engineering Mathematics- 3 - Question 1

Sum of the eigen values of matrix is = Sum of diagonal values present in the matrix

∴ 1 + 0 + P = 3 + λ+ λ3

⇒ P + 1 = 3 + λ+ λ3

⇒ λ+ λ= P + 1 – 3 = P – 2

Test: Engineering Mathematics- 3 - Question 2

What is the determinant of matrix X if 4 and (2 + 7i) are the eigenvalues of X where i = √−1?

Detailed Solution for Test: Engineering Mathematics- 3 - Question 2

Two eigen value of X is 4 and (2 + 7i)

∴ (2 - 7i) (conjugate roots) must be the third root

Determinant of P = product of eigenvalues

Δ = 4 × (2 + 7i) × (2 - 7i) 

Δ = 212

Test: Engineering Mathematics- 3 - Question 3

In the given matrix one of the eigenvalues is 1. The eigenvectors corresponding to the eigenvalue 1 are

Detailed Solution for Test: Engineering Mathematics- 3 - Question 3

For a given matrix A if V is the eigen vector corresponding to the eigen value λ, then:

AV = λV

∴ {α (−4, 2, 1) |α ≠ 0, αϵR} are the corresponding eigenvectors.

Test: Engineering Mathematics- 3 - Question 4

Which of the below-given statements is/are true?

I. The eigenvalue of the lower triangular matrix is just the diagonal elements of the matrix.

II. The product of the eigenvalue of a matrix is equal to its trace.

III. If 1/λ is an eigenvalue of A’(inverse of A) then orthogonal of A also have 1/λ as its eigenvalue.

Detailed Solution for Test: Engineering Mathematics- 3 - Question 4
  • The eigenvalue of the triangular (lower or upper) matrix are just the diagonal elements of matrix.
  • The product of the eigenvalue of a matrix is equal to its determinants.
  • If λ is the eigenvalue of matrix, then 1/λ is the eigenvalue of its inverse since orthogonal is equal to inverse matrix then it has 1/λ as its eigenvalue


​Example:

Eigenvalues are 1, 4 and 6 (diagonal elements)

Product of eigen value = determinants = 1 × 4 × 6 = 24

Orthogonal matrix and Inverse of given matrix have eigenvalues: 1,1/4 and 1/6

Test: Engineering Mathematics- 3 - Question 5

Consider the following 2 × 2 matrix A where two elements are unknown and are marked by a and b. The eigenvalues of this matrix are - 1 and 7. What are the values of a and b?

Detailed Solution for Test: Engineering Mathematics- 3 - Question 5

W.K.T.∑λ= ∑aii
∴ - 1 + 7 = 1 + a ⇒ a = 5

Also π λi = |A|

∴ - 1 × 7 = a – 4b

 - 7 = 5 – 4b ⇒ b = 3

Test: Engineering Mathematics- 3 - Question 6

Let A be the 2 X 2 matrix with elements a11 = 2, a12 = 3, a21 = 1 and a22 = 4 then the eigenvalues of the Matrix are A5?

Detailed Solution for Test: Engineering Mathematics- 3 - Question 6

Given Matrix:

Characteristic Equation:

(2−λ) (4−λ) −3 = 0

λ2 − 6λ + 5 = 0

λ = 5 or λ = 1

*Answer can only contain numeric values
Test: Engineering Mathematics- 3 - Question 7

The latent values of the matrix
 are 1, 1, 2 and the number of linearly independent latent vectors for the repeated root 1 is –


Detailed Solution for Test: Engineering Mathematics- 3 - Question 7

Eigen vectors are also called invariant vectors, characteristic vectors, or latent vectors. Eigenvalues are also called characteristic roots or latent roots.

(A - λI)X = 0

Given λ = 1

⇒ (A - I) X = 0

⇒ x = 2y + 3x = 0, 10x – 5y + 5z = 0, 5x – 4y + 5z = 0

By solving above equations, we get x = 

 i.e. one linearly independent latent vector.

*Answer can only contain numeric values
Test: Engineering Mathematics- 3 - Question 8

Consider a Matrix M = uTvT where u =  (112) and v = also uT denotes the transpose of matrix u. Find the largest eigenvalue of M?


Detailed Solution for Test: Engineering Mathematics- 3 - Question 8


Characteristic equation is given

 

λ3 − 5λ2 = 0

λ = 0 or λ = 5

Therefore largest value is 5.

Test: Engineering Mathematics- 3 - Question 9

What is the absolute difference of the eigenvalues for the matrix ad – bc = 6 and a + d = 7?

Detailed Solution for Test: Engineering Mathematics- 3 - Question 9

Let λ1 and λ2 be the two eigen values

Product of eigen value is equal to determinant of matrix

Sum of eigen value is equal to its trace
 

a + d = λ1 + λ2 = 7

λ1 × (7 − λ1) = 6

λ12 − 7λ + 6 = 0

λ1 = 6 or λ2 = 1

|λ1 − λ2| = 5

Test: Engineering Mathematics- 3 - Question 10

Consider the matrix  which one of the following statements is TRUE for the eigenvalues and eigenvectors of the matrix?

Detailed Solution for Test: Engineering Mathematics- 3 - Question 10


 

(5 - λ) (1 - λ) + 4 = 0

5 – 5λ - λ + λ2 + 4 = 0

λ2 - 6λ + 9 = 0

(λ - 3)2 = 0

λ = 3

for eigen vector


So, only one independent eigen vector.

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