Q$1.$ Eigenvalues and eigenvectors of a matrix

Consider the following matrix

$\backslash [$

$\backslash$left$[\backslash$begin$\{$array$\}\{$ccc$\}$

$2$ & $8$ & $10\backslash \backslash$

$8$ & $4$ & $5\backslash \backslash$

$10$ & $5$ & $7$

$\backslash$end$\{$array$\}\backslash$right$]$

$\backslash ]$

$($a$)(2$ points$)$ To find eigenvalues of this matrix using the polynomial method, derive the characteristic equation $($i$.$e$.,$ the equation that takes the form of polynomial $\backslash (=0\backslash )).$

$($b$)(2$ points$)$ To solve the characteristic equation, use a root finding code $($any method$)$ you previously wrote. You need to find at least one root $($i$.$e$.,$ one eigenvalue for the linear system$).$ Submit both your code and your result.

$($c$)(2$ point$)$ Find all eigenvalues using the same code and clearly explain how you achieved that.

$($d$)(4$ points$)$ Independently, write your own code to use the Power Method to determine both the largest and the smallest eigenvalues of the matrix.

Carry out $5$ iterations for each eigenvalue.

Calculate the approximate relative error at each iteration.

Note that you code should be able to handle a general square matrix of any size

$($Hint $01$: Refer to Examples $27\mathrm{.}7$ and $27\mathrm{.}8$ in the textbook.$).$

$($Hint $02$: You can use built$-$in functions in Matlab or Python to find the matrix inverse.$)$

$($e$)(2$ points$)$ Ask ChatGPT or similar Artificial Intelligence $($AI$)$ tools available online $($BingChat$,$ Bard, Claude, etc$)$ to write a piece of code for you with the same requirements specified in $($d$).$

Test the code and write a short $(1$ to $3$ sentences$)$ compare and contrast between your own code and code generated by the AI tool.

$($f$)($Extra Credit $2$ points$)$ Try to improve your own code so that it is at least better than the AI code in one aspect $($any aspect would be fine$).$

Clearly define this particular aspect you choose in $1$ sentence and then explain why your code is better. NOTE: Dr$.$ Pai and the TA team will not answer any questions regarding this extra credit task.

Please for the coding parts do it in MATLAB