Question

find the eigen values of the matrix 4 1 1 4​

Answer 1

Answer:

the eigen values of the matrix 4 1 1 4

-4114

-4411

-1144

Answer 2

The eigen values of the matrix are 3 & 5

Given :

The matrix

$\displaystyle\begin{pmatrix} 4 & 1 \\ 1 & 4 \end{pmatrix}$

To find :

The eigen values of the matrix

Solution :

Step 1 of 3 :

Write down the matrix

The given matrix is

$\displaystyle\begin{pmatrix} 4 & 1 \\ 1 & 4 \end{pmatrix}$

Step 2 of 3 :

Find the characteristic equation

The characteristic equation is given by

$\displaystyle\begin{vmatrix} 4 - \lambda & 1 \\ 1 & 4 - \lambda \end{vmatrix} = 0$

$\displaystyle \sf{ \implies {(4 - \lambda)}^{2} - 1 = 0}$

$\displaystyle \sf{ \implies {\lambda}^{2} - 8\lambda + 16 - 1 = 0}$

$\displaystyle \sf{ \implies {\lambda}^{2} - 8\lambda + 15 = 0}$

$\displaystyle \sf{ \implies {\lambda}^{2} - 8\lambda + 15 = 0}$

Step 3 of 3 :

Find eigen values of the matrix

$\displaystyle \sf{ {\lambda}^{2} - 8\lambda + 15 = 0}$

$\displaystyle \sf{ \implies {\lambda}^{2} - (5 + 3)\lambda + 15 = 0}$

$\displaystyle \sf{ \implies {\lambda}^{2} - 5\lambda - 3\lambda + 15 = 0}$

$\displaystyle \sf{ \implies\lambda( \lambda - 5) - 3(\lambda - 5)= 0}$

$\displaystyle \sf{ \implies( \lambda - 3) (\lambda - 5)= 0}$

$\displaystyle \sf{ \implies \lambda = 3 \: , \: 5}$

The eigen values of the matrix are 3 & 5