Determine the concentration C’s using the Matrix decomposition Crout’s/Cholesky’s method and Doolittle method...

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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 ⌋

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