1. Find LU Decomposition using Gauss Elimination method of Matrix ...
`[[3,2,4],[2,0,2],[4,2,3]]`

Solution:

Here `A`

=

`3` `2` `4` `2` `0` `2` `4` `2` `3`

Using Gaussian Elimination method
`R_2 larr R_2-``(2/3)``xx R_1` `[:.L_(2,1)=color{blue}{2/3}]`

=	`3` `2` `4` `0` `-4/3` `-2/3` `4` `2` `3`

`R_3 larr R_3-``(4/3)``xx R_1` `[:.L_(3,1)=color{blue}{4/3}]`

=	`3` `2` `4` `0` `-4/3` `-2/3` `0` `-2/3` `-7/3`

`R_3 larr R_3-``(1/2)``xx R_2` `[:.L_(3,2)=color{blue}{1/2}]`

=	`3` `2` `4` `0` `-4/3` `-2/3` `0` `0` `-2`

`:.U`

=

`3` `2` `4` `0` `-4/3` `-2/3` `0` `0` `-2`

`L` is just made up of the multipliers we used in Gaussian elimination with 1s on the diagonal.

`:.L`

=

`1` `0` `0` `color{blue}{2/3}` `1` `0` `color{blue}{4/3}` `color{blue}{1/2}` `1`

`:.` LU decomposition for A is

`A`

=

`3` `2` `4` `2` `0` `2` `4` `2` `3`

=

`1` `0` `0` `2/3` `1` `0` `4/3` `1/2` `1`

`xx`

`3` `2` `4` `0` `-4/3` `-2/3` `0` `0` `-2`

=

`LU`