Problem $2$:

Matrix inversion requires forward substitution and Back substitution after the decomposition of coefficient matrix A into $[$L$]$ and $[$U$].$ Show hand calculations of the third column of $A^{-1}$ and verify it by MATLAB.

Given that,

$[U]=\left[\begin{array}{ccc}1& \frac{1}{2}& \frac{1}{3}\\ 0& \frac{1}{6}& \frac{1}{6}\\ 0& 0& \frac{1}{60}\end{array}\right],[L]=\left[\begin{array}{ccc}1& 0& 0\\ 1& 1& 0\\ 1& \frac{3}{2}& 1\end{array}\right]$

$,[C]=$ Third column of the identity matrix

Solve$[L][Z]=[C]$ for $Z$