Solve Equations x+y+z=3,2x-y-z=3,x-y+z=9 using LU decomposition using Gauss Elimination method

Solution:
Total Equations are `3`

`x+y+z=3 -> (1)`

`2x-y-z=3 -> (2)`

`x-y+z=9 -> (3)`

Now converting given equations into matrix form
`[[1,1,1],[2,-1,-1],[1,-1,1]] [[x],[y],[z]]=[[3],[3],[9]]`

Now, A = `[[1,1,1],[2,-1,-1],[1,-1,1]]`, X = `[[x],[y],[z]]` and B = `[[3],[3],[9]]`

Here `A`

=

`1` `1` `1` `2` `-1` `-1` `1` `-1` `1`

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

=	`1` `1` `1` `0` `-3` `-3` `1` `-1` `1`

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

=	`1` `1` `1` `0` `-3` `-3` `0` `-2` `0`

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

=	`1` `1` `1` `0` `-3` `-3` `0` `0` `2`

`:.U`

=

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

`:.` LU decomposition for A is

`A`

=

`1` `1` `1` `2` `-1` `-1` `1` `-1` `1`

=

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

`xx`

`1` `1` `1` `0` `-3` `-3` `0` `0` `2`

=

`LU`

Now, `Ax=B`, and `A=LU => LUx=B`

let `Ux=y`, then `Ly=B =>`

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

`xx`

`y_1` `y_2` `y_3`

=

`3` `3` `9`

			``	`y_1`					`=`	`3`	``
			``	`2y_1`	`+`	`y_2`			`=`	`3`	``
			``	`y_1`	`+`	`2/3y_2`	`+`	`y_3`	`=`	`9`	``

Now use forward substitution method
From (1)
`y_1=3`

From (2)
`2y_1+y_2=3`

`=>2(3)+y_2=3`

`=>6+y_2=3`

`=>y_2=3-6`

`=>y_2=-3`

From (3)
`y_1+2/3y_2+y_3=9`

`=>(3)+(2(-3))/(3)+y_3=9`

`=>1+y_3=9`

`=>y_3=9-1`

`=>y_3=8`

Now, `Ux=y`

`1` `1` `1` `0` `-3` `-3` `0` `0` `2`

`xx`

`x` `y` `z`

=

`3` `-3` `8`

			``	`x`	`+`	`y`	`+`	`z`	`=`	`3`	``
					`-`	`3y`	`-`	`3z`	`=`	`-3`	``
							``	`2z`	`=`	`8`	``

Now use back substitution method
From (3)
`2z=8`

`=>z=(8)/(2)=4`

From (2)
`-3y-3z=-3`

`=>-3y-3(4)=-3`

`=>-3y-12=-3`

`=>-3y=-3+12`

`=>-3y=9`

`=>y=(9)/(-3)=-3`

From (1)
`x+y+z=3`

`=>x+(-3)+(4)=3`

`=>x+1=3`

`=>x=3-1`

`=>x=2`

Solution by LU decomposition method is
`x=2,y=-3 and z=4`

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