DOLITTLE METHOD
A=
⌈1 7 -4 ⌉
| 4 -4 9 | (C matrix)
⌊12 -1 3 ⌋
B=
⌈ 6 ⌉
| 39 | (constants)
⌊26 ⌋
C=
⌈C₁ ⌉
| C₂ | (C’s)
⌊C₃ ⌋
L=
⌈1 0 0 ⌉
| l₂₁ 1 0 | (lower triangular matrix)
⌊l₃₁ l₃₂ 1 ⌋
U=
⌈u₁₁ u₁₂ u₁₃ ⌉
| 0 u₂₂ u₂₃ | (upper triangular matrix)
⌊ 0 0 u₃₃ ⌋
Apply A=LU, equating elements using matrix multiplication:
L=
⌈1 0 0 ⌉
| 4 1 0 |
⌊12 85/32 1 ⌋
U=
⌈1 7 -4 ⌉
| 0 -32 25 |
⌊ 0 0 -493/32 ⌋
Check this out by matrix multiplication: L×U=A.
Introduce matrix Y=
⌈y₁ ⌉
| y₂ |
⌊y₃ ⌋
Find Y using LY=B, equating elements:
Y=
⌈ 6 ⌉
| 15 |
⌊ -2747/32 ⌋
Check this out by matrix multiplication: L×Y=B.
Apply UC=Y to find C, equating elements using matrix multiplication:
C=
⌈ 541/493 ⌉
| 1915/493 |
⌊2747/493 ⌋
So the concentrations are:
C₁=541/493=1.0974 g/m³
C₂=1915/493=3.8844 g/m³
C₃=2747/493=5.5720 g/m³
CROUT METHOD is similar but the diagonal 1s are in the upper triangular matrix, instead of the lower one:
L=
⌈1 0 0 ⌉
| 4 -32 0 |
⌊12 -85 -493/32 ⌋
U=
⌈1 7 -4 ⌉
| 0 1 -25/32 |
⌊ 0 0 1 ⌋
Y=
⌈ 6 ⌉
| -15/32 |
⌊ 2747/493 ⌋