# Eigenvalues and Eigenvectors

Suppose *A* is a square matrix. The relation for finding Eigenvalue _{} corresponds to the Eigenvector **x** is

A**x** = _{}**x**

# Practice Problems

**Find Eigenvalue corresponds to the Eigenvector for matrix**

*Example 1.*

*Solution*As we know that A

**x**=

_{}

**x**

So,

Hence,

_{}= 7

*Answer*# How to find Eigenvalues of a matrix

To find eigenvalues of a given matrix, we have to use the relationdet ( _{} *I* - *A* ) = 0

**Find all the eigenvalues of the matrix**

*Example 2.*

*Solution*As we know that det (

_{}

*I*-

*A*) = 0

Now,

Now find determinant of

_{}

*I*-

*A*

det (

_{}

*I*-

*A*) = (

_{}- 4 ) (

_{}- 11 ) - ( -3 ) ( 4 )

=

_{}

^{2}- 11

_{}- 4

_{}+ 44 + 12

=

_{}

^{2}- 15

_{}+ 56

Now set this equal to zero to obtain det (

_{}

*I*-

*A*) = 0 and solve for the eigenvalues.

_{}

^{2}- 15

_{}+ 56 = 0

_{}

^{2}- 8

_{}- 7

_{}+ 56 = 0

_{}(

_{}- 8 ) - 7 (

_{}- 8 ) = 0

(

_{}- 7 ) (

_{}- 8 ) = 0

_{}

_{1}= 7, and

_{2}= 8

So we get two eigenvalues.

*Answer*