# GATE Questions & Answers of Eigen Values and Eigen Vectors

## What is the Weightage of Eigen Values and Eigen Vectors in GATE Exam?

Total 15 Questions have been asked from Eigen Values and Eigen Vectors topic of Linear Algebra subject in previous GATE papers. Average marks 1.40.

The product of eigenvalues of matrix P is

$\mathrm P=\begin{bmatrix}2&0&1\\4&-3&3\\0&2&-1\end{bmatrix}$

Consider the matrix $\style{font-family:'Times New Roman'}{\mathrm P=\begin{bmatrix}\frac1{\sqrt2}&0&\frac1{\sqrt2}\\0&1&0\\\frac{-1}{\sqrt2}&0&\frac1{\sqrt2}\end{bmatrix}}$

Which one of the following statements about P is INCORRECT?

The determinant of a 2x2 matrix is 50. If one eigenvalue of the matrix is 10, the other eigenvalue is_________

Consider the matrix A=$\left[\begin{array}{cc}50& 70\\ 70& 80\end{array}\right]$ whose eigenvectors corresponding to eigenvalues ${\lambda }_{1}$ and ${\lambda }_{2}$ are ${\mathrm{X}}_{1}=\left[\begin{array}{cc}70& \\ {\lambda }_{1}& -50\end{array}\right]$ and ${\mathrm{X}}_{2}=\left[\begin{array}{cc}{\lambda }_{2}& -80\\ 70& \phantom{\rule{0ex}{0ex}}\end{array}\right]$ , respectively. The value of x1T x2 is___________

One of the eigenvectors of the matrix $\left[\begin{array}{cc}-5& 2\\ -9& 6\end{array}\right]$

Consider a 3×3 real symmetric matrix S such that two of its eigenvalues are $a\ne 0,b\ne 0$ with respective eigenvectors $\left[\begin{array}{c}{x}_{1}\\ {x}_{2}\\ {x}_{3}\end{array}\right],$$\left[\begin{array}{c}{y}_{1}\\ {y}_{2}\\ {y}_{3}\end{array}\right]$.If $a\ne b$ then ${x}_{1}{y}_{1}+{x}_{2}{y}_{2}+{x}_{3}{y}_{3}$ equals

The eigenvalues of a symmetric matrix are all

For the matrix $A=\left[\begin{array}{cc}5& 3\\ 1& 3\end{array}\right]$, ONE of the normalized eigen vectors is given as

Eigenvalues of a real symmetric matrix are always

One of the eigen vectors of the matrix $A=\left[\begin{array}{cc}2& 2\\ 1& 3\end{array}\right]$ is

For a matrix [M]= $\left[\begin{array}{cc}\frac{3}{5}& \frac{4}{5}\\ x& \frac{3}{5}\end{array}\right]$, the transpose of the matrix is equal to the inverse of the matrix [M]T = [M]-1. The value of x is given by

The matrix  $\left[\begin{array}{ccc}1& 2& 4\\ 3& 0& 6\\ 1& 1& p\end{array}\right]$ has one eigen value equal to 3. The sum of the other two eigenvalues is

The eigenvectors of the matrix $\left[\begin{array}{cc}1& 2\\ 0& 2\end{array}\right]$ are written in the form$\left[\begin{array}{c}1\\ a\end{array}\right]$ and $\left[\begin{array}{c}1\\ b\end{array}\right]$ What is a + b?

The number of linearly independent eigenvectors of $\left[\begin{array}{cc}2& 1\\ 0& 2\end{array}\right]$ is