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Test: Eigenvalues & Eigenvectors - 2 - Civil Engineering (CE) MCQ


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25 Questions MCQ Test Engineering Mathematics - Test: Eigenvalues & Eigenvectors - 2

Test: Eigenvalues & Eigenvectors - 2 for Civil Engineering (CE) 2024 is part of Engineering Mathematics 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) 2024 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

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 7

The eigen values of the matrix  

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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

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 10

If    then top row of R-1 is 

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Test: Eigenvalues & Eigenvectors - 2 - Question 11

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  


=> (λ + 1)(λ +2)
By Cayley theorem (A + 1I2 ) ( A + 2I2 )

Test: Eigenvalues & Eigenvectors - 2 - Question 12

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 13

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 14

An eigenvector of 

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Eigen values of P are 1,2,3 

Test: Eigenvalues & Eigenvectors - 2 - Question 15

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 16

Detailed Solution for Test: Eigenvalues & Eigenvectors - 2 - Question 16

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

As diagonal elements are eigen valves.

Test: Eigenvalues & Eigenvectors - 2 - Question 17

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 18

The eigenvalues of

are

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

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 19

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 19

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 20

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 

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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 21

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 22

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 23

The condition for which eigenvalues of the matrix A are positive, is

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Suppose λ1 and λ2 are eigenvalues of matrix A.

λ1λ2 = |A| = 2k - 1>0

λ1 + λ2 = 2+k >0

k>1/2 and k>-2

Both the conditions will be true for k>1/2

Hence option A is the correct answer.

Test: Eigenvalues & Eigenvectors - 2 - Question 24

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

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Test: Eigenvalues & Eigenvectors - 2 - Question 25

Consider a 3 × 3 real symmetric matrix S such that two of its eigen values are a ≠ 0 , b ≠ 0 with respective eigenvectors X and Y. If a ≠ b then x1y1 + x2y2 + x3y3 equals

Detailed Solution for Test: Eigenvalues & Eigenvectors - 2 - Question 25

For symmetric matrix having two distinct eigenvalues, corresponding eigenvectors are orthogonal.

x1y1 + x2y2 + x3y3 = 0

Hence option D is the correct answer.

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