# GATE Questions & Answers of Eigen Values and Eigen vectors

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

Total 9 Questions have been asked from Eigen Values and Eigen vectors topic of Linear Algebra subject in previous GATE papers. Average marks 1.67.

A system matrix is given as follows

$A=\left[\begin{array}{ccc}0& 1& -1\\ -6& -11& 6\\ -6& -11& 5\end{array}\right]$

The absolute value of the ratio of the maximum eigenvalue to the minimum eigenvalue is _______

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Which one of the following statements is true for all real symmetric matrices?

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A matrix has eigenvalues –1 and –2. The corresponding eigenvectors are $\left[\begin{array}{c}1\\ -1\end{array}\right]$ and $\left[\begin{array}{c}1\\ -2\end{array}\right]$ respectively. The matrix is

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An eigenvector of $\mathrm{p}=\left(\begin{array}{ccc}1& 1& 0\\ 0& 2& 2\\ 0& 0& 3\end{array}\right)$ is

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The trace and determinant of a 2 × 2 matrix are known to be -2 and -35 respectively. Its eigen values are

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The characteristic equation of a (3X3) matrix P is defined as

$\alpha \left(\lambda \right)=|\lambda \mathbf{I}-\mathbf{P}|={\lambda }^{3}+{\lambda }^{2}+1=0$

If I denotes identity matrix, then the inverse of  matrix P will be

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The linear operation L(x) is defined by the cross product L(x) = bXx, where b=[0 1 0]T and x=[x1 x2 x3]T are three dimensional vectors. The 3×3 matrix M of this operations satisfies $\mathrm{L}\left(\mathrm{x}\right)=\mathrm{M}\left[\begin{array}{c}{x}_{1}\\ {x}_{2}\\ {x}_{3}\end{array}\right]$

Then the eigenvalues of M are

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Cayley-Hamilton Theorem states that a square matrix satisfies its own characteristic equation. Consider a matrix $A=\left[\begin{array}{cc}-3& 2\\ -1& 0\end{array}\right]$

A satisfies the relation

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Cayley-Hamilton Theorem states that a square matrix satisfies its own characteristic equation. Consider a matrix $A=\left[\begin{array}{cc}-3& 2\\ -1& 0\end{array}\right]$

A9 equals