Problem $1$

Solve the following set of equations using LU decomposition:$123$

$123$

$123$

$8411$

$254$

$267$

x x x

x x x

x x x

$+=$

$++=$

$+=$

Problem $2$

Determine the LU decomposition for the following matrix$\mathrm{.}821$

$372$

$239$

$$

$$

$$

$$

$($a$)$ Check your results by validating that $[$L$][$U$]=[$A$].$

$($b$)$ Employ the result of part $($a$)$ to compute the determinant.

Problem $3$

Use the Gauss$-$Seidel method $($a$)$ without relaxation and $($b$)$ with relaxation $($$=0\mathrm{.}95)$ to solve

the following system to a tolerance of $$s $=5\%.$ If necessary, rearrange the equations to achieve

convergence$\mathrm{.}123$

$123$

$123$

$31250$

$63$

$6940$

x x x

x x x

x x x

$++=$

$=$

$++=$