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LU decomposition using Doolittle's method of Matrix calculator
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 1. `[[8,-6,2],[-6,7,-4],[2,-4,3]]`
2. `[[6,-2,2],[-2,3,-1],[2,-1,3]]`
3. `[[3,2,4],[2,0,2],[4,2,3]]`
4. `[[1,1,1],[-1,-3,-3],[2,4,4]]`
5. `[[2,3],[4,10]]`
6. `[[5,1],[4,2]]`
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Example1. Find LU decomposition using Doolittle's method of Matrix ...
`[[8,-6,2],[-6,7,-4],[2,-4,3]]`Solution:Doolittle's method for LU decomposition Let `A=LU` | `8` | `-6` | `2` | | | `-6` | `7` | `-4` | | | `2` | `-4` | `3` | |
| = | | `1` | `0` | `0` | | | `l_(21)` | `1` | `0` | | | `l_(31)` | `l_(32)` | `1` | |
| `xx` | | `u_(11)` | `u_(12)` | `u_(13)` | | | `0` | `u_(22)` | `u_(23)` | | | `0` | `0` | `u_(33)` | |
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| `8` | `-6` | `2` | | | `-6` | `7` | `-4` | | | `2` | `-4` | `3` | |
| = | | `u_(11)` | `u_(12)` | `u_(13)` | | | `l_(21)u_(11)` | `l_(21)u_(12) + u_(22)` | `l_(21)u_(13) + u_(23)` | | | `l_(31)u_(11)` | `l_(31)u_(12) + l_(32)u_(22)` | `l_(31)u_(13) + l_(32)u_(23) + u_(33)` | |
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This implies `u_(11)=8` `u_(12)=-6` `u_(13)=2` `l_(21)u_(11)=-6=>l_(21)xx8=-6=>l_(21)=-3/4` `l_(21)u_(12) + u_(22)=7=>(-3/4)xx(-6) + u_(22)=7=>u_(22)=5/2` `l_(21)u_(13) + u_(23)=-4=>(-3/4)xx2 + u_(23)=-4=>u_(23)=-5/2` `l_(31)u_(11)=2=>l_(31)xx8=2=>l_(31)=1/4` `l_(31)u_(12) + l_(32)u_(22)=-4=>1/4xx(-6) + l_(32)xx5/2=-4=>l_(32)=-1` `l_(31)u_(13) + l_(32)u_(23) + u_(33)=3=>1/4xx2 + (-1)xx(-5/2) + u_(33)=3=>u_(33)=0` `:.A=L xx U=LU` | `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` | |
| = | | `8` | `-6` | `2` | | | `-6` | `7` | `-4` | | | `2` | `-4` | `3` | |
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