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 ⌋