# GATE Questions & Answers of Eigen values and Eigen vectors

## What is the Weightage of Eigen values and Eigen vectors in GATE Exam?

Total 14 Questions have been asked from Eigen values and Eigen vectors topic of Linear Algebra subject in previous GATE papers. Average marks 1.64.

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 is .

Consider a matrix $\style{font-family:'Times New Roman'}{A=uv^T\;}$ where $\style{font-family:'Times New Roman'}{u=\begin{pmatrix}1\\2\end{pmatrix},\;v\;=\begin{pmatrix}1\\1\end{pmatrix}}$ . Note that $\style{font-family:'Times New Roman'}{v^T}$ denotes the transpose of $v$. The largest eigenvalue of $A$ is _____.

Consider a matrix P whose only eigenvectors are the multiples of $\style{font-family:'Times New Roman'}{\begin{bmatrix}1\\4\end{bmatrix}}$.

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?

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

$\left[\begin{array}{ccccc}1& 0& 0& 0& 1\\ 0& 1& 1& 1& 0\\ 0& 1& 1& 1& 0\\ 0& 1& 1& 1& 0\\ 1& 0& 0& 0& 1\end{array}\right]$

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.

$\left[\begin{array}{ccc}1& 2& 3\\ 0& 4& 7\\ 0& 0& 3\end{array}\right]$

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

Consider the following matrix

$A=\left[\begin{array}{cc}2& 3\\ x& y\end{array}\right]$

If the eigenvalues of A are 4 and 8, then

Let A be a 4 × 4 matrix with eigenvalues -5, -2, 1, 4. Which of the following is an eigenvalue of $\left[\begin{array}{cc}A& I\\ I& A\end{array}\right]$, where I is the 4 × 4 identity matrix?