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Find lu decomposition [[8,-6,2],[-6,7,-4],[2,-4,3]]

Solution:
Your problem `->` lu decomposition [[8,-6,2],[-6,7,-4],[2,-4,3]]


`LU` decomposition : If we have a matrix A, then an upper triangular matrix U can be obtained without pivoting under Gaussian Elimination method, and there exists lower triangular matrix L such that A=LU.


Here `A` = 
`8``-6``2`
`-6``7``-4`
`2``-4``3`


Using Gaussian Elimination method
`R_2 larr R_2-` `(-3/4)` `-3/4=(-6)/(8)``xx R_1` `[:.L_(2,1)=color{blue}{-3/4}]`

 = 
`8``-6``2`
 `0` `0=-6-(-3/4)xx8`
`R_2 larr R_2-(-3/4)xx R_1`
 `5/2` `5/2=7-(-3/4)xx-6`
`R_2 larr R_2-(-3/4)xx R_1`
 `-5/2` `-5/2=-4-(-3/4)xx2`
`R_2 larr R_2-(-3/4)xx R_1`
`2``-4``3`


`R_3 larr R_3-` `(1/4)` `1/4=(2)/(8)``xx R_1` `[:.L_(3,1)=color{blue}{1/4}]`

 = 
`8``-6``2`
`0``5/2``-5/2`
 `0` `0=2-(1/4)xx8`
`R_3 larr R_3-(1/4)xx R_1`
 `-5/2` `-5/2=-4-(1/4)xx-6`
`R_3 larr R_3-(1/4)xx R_1`
 `5/2` `5/2=3-(1/4)xx2`
`R_3 larr R_3-(1/4)xx R_1`


`R_3 larr R_3-` `(-1)` `-1=(-5/2)/(5/2)``xx R_2` `[:.L_(3,2)=color{blue}{-1}]`

 = 
`8``-6``2`
`0``5/2``-5/2`
 `0` `0=0-(-1)xx0`
`R_3 larr R_3-(-1)xx R_2`
 `0` `0=-5/2-(-1)xx5/2`
`R_3 larr R_3-(-1)xx R_2`
 `0` `0=5/2-(-1)xx-5/2`
`R_3 larr R_3-(-1)xx R_2`


`:.U` = 
`8``-6``2`
`0``5/2``-5/2`
`0``0``0`


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

`:.L` = 
`1``0``0`
`color{blue}{-3/4}``1``0`
`color{blue}{1/4}``color{blue}{-1}``1`


`:.` LU decomposition for A is

`A` = 
`8``-6``2`
`-6``7``-4`
`2``-4``3`
 = 
`1``0``0`
`-3/4``1``0`
`1/4``-1``1`
 `xx` 
`8``-6``2`
`0``5/2``-5/2`
`0``0``0`
 = `LU`







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