Question: $[20$ Marks$]$

Consider the following linear system

$3+24=9$

$735+2=6$

$+++=10$

$2+53=3$

a$)$ Solve the system using Matlab$/$Octave rref, backslash, and the inverse commands. Call the

solution X$.[3]$

b$)$ Compute the spectral radius of the coefficient matrix. $[2]$

c$)$ Use the $2-$norm to Compute the condition number of the coefficient matrix. What the

number tells you about the matrix. $[2]$

d$)$ Use Matlab$/$Octave code to solve the system using the Jacobi method starting with initial

value $$

$(0)=(0\mathrm{.}5,0\mathrm{.}5,0\mathrm{.}5,0\mathrm{.}5,0\mathrm{.}5).$ Call the solution $.$ Print $10$ iterations. $[4]$

e$)$ Use Matlab$/$Octave code to solve the system using the Gauss$-$Seidel method starting with initial

value $$

$(0)=(0\mathrm{.}5,0\mathrm{.}5,0\mathrm{.}5,0\mathrm{.}5,0\mathrm{.}5).$ Call the solution $.$ Print $10$ iterations. $[4]$

f$)$ Use Matlab$/$Octave code to compute the $2$ and $2$for the errors in the

approximation. $[3]$

g$)$ Are the two numerical methods converging to the solution? Which method is better to use.

Justify your answer.