GATE CS Eigenvalues and Eigenvectors, LU Decomposition questions

Practise Eigenvalues and Eigenvectors, LU Decomposition questions for GATE CS. Explore official PYQs and original topic-wise quizzes, check answers and view available solutions. Free, no sign-in.

20 on this page

SubjectTopicYearType

n>1

, the maximum multiplicity of any eigenvalue of an

n \times n

matrix with elements from

\mathbb{R}

A

2 \times 2

matrix as given.

A=\left[\begin{array}{rr} 1 & 1 \\ 1 & -1 \end{array}\right]

What are the eigenvalues of the matrix

A^{13}

The product of all eigenvalues of the matrix

\left[\begin{array}{lll}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9\end{array}\right]

A

be the adjacency matrix of the graph with vertices

\{1,2,3,4,5\}.

\lambda_{1}, \lambda_{2}, \lambda_{3}, \lambda_{4}

\lambda_{5}

be the five eigenvalues of

A

. Note that these eigenvalues need not be distinct. The value of

\lambda_{1}+\lambda_{2}+\lambda_{3}+\lambda_{4}+\lambda_{5}=

Which of the following is/are the eigenvector(s) for the matrix given below?

\begin{pmatrix} – 9 & – 6 & – 2 & – 4 \\ – 8 & – 6 & – 3 & – 1 \\ 20 & 15 & 8 & 5 \\ 32 & 21 & 7 & 12 \end{pmatrix}

Consider the following matrix.

\begin{pmatrix} 0 & 1 & 1 & 1\\ 1& 0& 1 & 1\\ 1& 1 & 0 & 1 \\1 & 1 & 1 & 0 \end{pmatrix}

The largest eigenvalue of the above matrix is __________.

Consider the following matrix:

R = \begin{bmatrix} 1 & 2 & 4 & 8 \\ 1 & 3 & 9 & 27 \\ 1 & 4 & 16 & 64 \\ 1 & 5 & 25 & 125 \end{bmatrix}

The absolute value of the product of Eigen values of

R

Consider a matrix P whose only eigenvectors are the multiples of

\begin{bmatrix} 1 \\ 4 \end{bmatrix}

. Consider the following statements. P does not have an inverse P has a repeated eigenvalue P cannot be diagonalized Which one of the following options is correct?

Consider a matrix

A= uv^T

u=\begin{pmatrix}1 \\ 2 \end{pmatrix} , v = \begin{pmatrix}1 \\1 \end{pmatrix}

v^T

denotes the transpose of

v

. The largest eigenvalue of

A

A

n\times n

real valued square symmetric matrix of rank

2

\sum_{i=1}^{n}\sum_{j=1}^{n}A^{2}_{ij} = 50.

Consider the following statements. One eigenvalue must be in

\left [ -5,5 \right ]

The eigenvalue with the largest magnitude must be strictly greater than

5

Which of the above statements about eigenvalues of

A

If the characteristic polynomial of a

3 \times 3

M

\mathbb{R}

(the set of real numbers) is

\lambda^3 – 4 \lambda^2 + a \lambda +30, \quad a \in \mathbb{R}

, and one eigenvalue of

M

2,

then the largest among the absolute values of the eigenvalues of

M

Two eigenvalues of a

3 \times 3

P

(2+\sqrt {-1})

3

. The determinant of

P

Suppose that the eigenvalues of matrix

A

1, 2, 4

. The determinant of

\left(A^{-1}\right)^{T}

Consider the following

2 \times 2

A

where two elements are unknown and are marked by

a

b

. The eigenvalues of this matrix are

-1

7.

What are the values of

a

b

\qquad A = \begin{pmatrix}1 & 4\\ b&a \end{pmatrix}

The larger of the two eigenvalues of the matrix

\begin{bmatrix} 4 & 5 \\ 2 & 1 \end{bmatrix}

In the given matrix

\begin{bmatrix} 1 & -1 & 2 \\ 0 & 1 & 0 \\ 1 & 2 & 1 \end{bmatrix}

, one of the eigenvalues is

1.

The eigenvectors corresponding to the eigenvalue

1

The value of the dot product of the eigenvectors corresponding to any pair of different eigenvalues of a

4-by-4

symmetric positive definite matrix is ___________

The product of the non-zero eigenvalues of the matrix is ____

\begin{pmatrix} 1 & 0 & 0 & 0 & 1 \\ 0 & 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 & 0 \\ 1 & 0 & 0 & 0 & 1 \end{pmatrix}

Which one of the following statements is TRUE about every

n \times n

matrix with only real eigenvalues?

2 × 2

matrix with elements

a_{11} = a_{12} = a_{21} = +1

a_{22} = −1

. Then the eigenvalues of the matrix

A^{19}