LU Decomposition using Gauss Elimination method of Matrix calculator

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Solution

Solution provided by AtoZmath.com

1. `[[8,-6,2],[-6,7,-4],[2,-4,3]]`

2. `[[3,2,4],[2,0,2],[4,2,3]]`

3. `[[1,1,1],[-1,-3,-3],[2,4,4]]`

4. `[[1,2,3],[0,1,0],[2,3,1]]`

5. `[[1,2,1],[6,-1,0],[-1,-2,-1]]`

6. `[[6,-2,2],[-2,3,-1],[2,-1,3]]`

7. `[[2,3],[4,10]]`

8. `[[5,1],[4,2]]`

9. `[[6,3],[4,5]]`

Example

1. Find LU Decomposition using Gauss Elimination method of Matrix ...
`[[8,-6,2],[-6,7,-4],[2,-4,3]]`

Solution:

`LU` decomposition : If we have a square 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-``(-0.75)``xx R_1` `[:.L_(2,1)=color{blue}{-0.75}]`

`8`	`-6`	`2`
`0`	`2.5`	`-2.5`
`2`	`-4`	`3`

`R_3 larr R_3-``(0.25)``xx R_1` `[:.L_(3,1)=color{blue}{0.25}]`

`8`	`-6`	`2`
`0`	`2.5`	`-2.5`
`0`	`-2.5`	`2.5`

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

`8`	`-6`	`2`
`0`	`2.5`	`-2.5`
`0`	`0`	`0`

`:.U`

`8`	`-6`	`2`
`0`	`2.5`	`-2.5`
`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}{-0.75}`	`1`	`0`
`color{blue}{0.25}`	`color{blue}{-1}`	`1`

Now checking `A=LU` ?

`LU`

`1`	`0`	`0`
`-0.75`	`1`	`0`
`0.25`	`-1`	`1`

`xx`

`8`	`-6`	`2`
`0`	`2.5`	`-2.5`
`0`	`0`	`0`

`8`	`-6`	`2`
`-6`	`7`	`-4`
`2`	`-4`	`3`

And `A`

`8`	`-6`	`2`
`-6`	`7`	`-4`
`2`	`-4`	`3`

Solution is possible.