Eigenvalues and Eigenvectors

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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

  • A

    1024 and -1024

  • B

    1024√2 and -1024√2

  • C

    4√2 and -4√2

  • D

    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?

  • A

    1, 4, 3

  • B

    3, 7, 3

  • C

    7, 3, 2

  • D

    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

 

  • A

    x=-3, y=9
     

  • B

    x= -4, y=10
     

  • C

    x=5, y=8
     

  • D

    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]

  • A

    Four

  • B

    Three

  • C

    Two

  • D

    One

Question 5

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

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

  • A

    -1 and 1

  • B

    1 and 6

  • C

    2 and 5

  • D

    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 ____

  • A

    5

  • B

    6

  • C

    7

  • D

    8

Question 7

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

  • A

    1/8

  • B

    1

  • C

    1/4

  • D

    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 __________ .

  • A

    84

  • B

    588

  • C

    49

  • D

    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?

  • A

    Only I and III are necessarily true

  • B

    Only II is necessarily true

  • C

    Only I and II are necessarily true

  • D

    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 __________.

  • A

    3

  • B

    4

  • C

    0

  • D

    12

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