Solve the following equation,

$\left[\begin{array}{ccccc}3& -1& 0& 0& 0\\ 2& 3& -1& 0& 0\\ 0& 2& 3& -1& 0\\ 0& 0& 2& 3& -1\\ 0& 0& 0& 2& 3\end{array}\right]x_{?}=\{[0\mathrm{.}1],[5],[3],[0],[5]\}$

$($a$)$ By Gaussian elimination method

$($b$)$ By Doolittle method

$($c$)$ By Cholesky method

$($d$)$ By Thomas method

$($e$)$ By Jacobi method

$($f$)$ By Gauss$-$Seidel method

For $($e$)$ and $($f$),$ use error tolerance, $={10}^{-6}$ and the initial $x^{(0)}$ vector,

$x_{?}={?}^{(0)}\{[1],[0],[1],[0],[1]\}$