Eigenvalues and Eigenvectors

Last Updated :
Discuss
Comments

Question 1

Let A be the 2 Γ— 2 matrix with elements a11 = a12 = a21 = +1 and a22 = βˆ’1. Then the eigenvalues of the matrix A19 are

  • 1024 and -1024

  • 1024√2 and -1024√2

  • 4√2 and -4√2

  • 512√2 and -512√2

Question 2

Consider the matrix as given below.

[Tex]\begin{bmatrix} 1&2&3\\ 0&4&7\\ 0&0&3\end{bmatrix}[/Tex]

Which one of the following options provides the CORRECT values of the eigenvalues of the matrix?

  • 1, 4, 3

  • 3, 7, 3

  • 7, 3, 2

  • 1, 2, 3

Question 3

Consider the following matrix 

[Tex]A = \begin{bmatrix} 2 & 3 \\ x & y \end{bmatrix}[/Tex]

If the eigenvalues of A are 4 and 8, then

 

  • x=-3, y=9
     

  • x= -4, y=10
     

  • x=5, y=8
     

  • x=4, y=10
     

Question 4

How many of the following matrices have an eigenvalue 1?

[Tex]\left[\begin{array}{ll}1 & 0 \\ 0 & 0\end{array}\right],\left[\begin{array}{ll}0 & 1 \\ 0 & 0 \end{array}\right],\left[\begin{array}{cc}1 & -1 \\1 & 1 \end{array}\right][/Tex] and [Tex]\left[\begin{array}{cc} -1 & 0 \\1 & -1 \end{array}\right][/Tex]

  • Four

  • Three

  • Two

  • One

Question 5

What are the eigenvalues of the following 2 Γ— 2 matrix?

[Tex]\begin{bmatrix} 2 & -1 \\ -4 & 5 \end{bmatrix}[/Tex]

  • -1 and 1

  • 1 and 6

  • 2 and 5

  • 4 and -1

Question 6

The larger of the two eigenvalues of the matrix [Tex]\begin{bmatrix} 4 & 5 \\ 2 & 1 \end{bmatrix}[/Tex] is ____

  • 5

  • 6

  • 7

  • 8

Question 7

Suppose that the eigenvalues of matrix A are 1, 2, 4. The determinant of (Aβˆ’1)T is _________

  • 1/8

  • 1

  • 1/4

  • 2

Question 8

The matrix A has (1, 2, 1)T and (1, 1, 0)T as eigenvectors, both with eigenvalue 7, and its trace is 2. The determinant of A is __________ .

  • 84

  • 588

  • 49

  • None of these

Question 9

Consider a matrix P(2x2) whose only eigenvectors are the multiples of [Tex]\begin{bmatrix}1 \\ 4 \end{bmatrix}[/Tex]
Consider the following statements.
(I) P does not have an inverse
(II) P has a repeated eigenvalue
(III) P cannot be diagonalized
Which one of the following options is correct?

  • Only I and III are necessarily true

  • Only II is necessarily true

  • Only I and II are necessarily true

  • Only II and III are necessarily true

Question 10

Consider the following matrix.

[Tex]\begin{pmatrix}0 & 1 & 1 & 1\\ 1 & 0 & 1 & 1 \\1 & 1 & 0 & 1 \\1 & 1 & 1 & 0\end{pmatrix}[/Tex]

The largest eigenvalue of the above matrix is __________.

  • 3

  • 4

  • 0

  • 12

Tags:
GATE CS

There are 10 questions to complete.

Take a part in the ongoing discussion