Questions & Answers of Eigen values and Eigen vectors

Weightage of Eigen values and Eigen vectors

Total 11 Questions have been asked from Eigen values and Eigen vectors topic of Linear Algebra subject in previous GATE papers. Average marks 1.64.

Let A be $\style{font-family:'Times New Roman'}{n\times n}$ real valued  square symmetric matrix of rank 2 with $\style{font-family:'Times New Roman'}{\sum_\limits{i=1}^n\sum_\limits{j=1}^nA_{ij}^2=50.}$ Consider the following statements.
 
(I) One eigenvalue must be in [-5, 5]
 
(II) The eigenvalue with the largest magnitude must be strictly greater than 5
 
Which of the above statments about eigenvalue of A is/are necessarily CORRECT?
 

Let $ P=\begin{bmatrix}1&1&-1\\2&-3&4\\3&-2&3\end{bmatrix} $ and $ Q=\begin{bmatrix}-1&-2&-1\\6&12&6\\5&10&5\end{bmatrix} $ be two matrices.

Then the rank of P+Q is ________.

If the characteristics polynomial of a $\style{font-family:'Times New Roman'}{3\times3}$ matrix M over $\mathbb{R}$ (the set of real numbers) is $\style{font-family:'Times New Roman'}{\lambda^3-4\lambda^2+a\lambda+30,\;a\in\mathbb{R}}$ and one eigenvalue of M is 2, then the largest among the absolute values of the eigenvalues of M is _____________. 

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

1000101110011100111010001

is ______.

Which one of the following statements is TRUE about every n × n matrix with only real eigenvalues?

Let A be the 2 × 2 matrix with elements a11 = a12 = a21 = +1 and a22 = −1. Then the eigenvalues of the matrix A19 are

Consider the matrix as given below.

123047003

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

Consider the following matrix

A=23xy

If the eigenvalues of A are 4 and 8, then

How many of the following matrices have an eigenvalue 1?

1000,0100,1-111  and -101-1

Let A be a 4 × 4 matrix with eigenvalues -5, -2, 1, 4. Which of the following is an eigenvalue of AIIA, where I is the 4 × 4 identity matrix?