 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2019/digital-systems-linear-algebra-introduction-to-formal-proof/4/c/ii/solution

Apply the same permutation to the right-hand side and solve two triangular systems. Forward substitution in

L y = P b

gives

y_{1} = 3, 2 y_{1} + y_{2} = 0, 3 y_{1} + y_{3} = - 2,

y = (3, - 6, - 11)^{T}

Ux = y

gives

- 11 x_{3} = - 11

4 x_{2} - 2 x_{3} = - 6

, and

3 x_{1} + x_{2} + 4 x_{3} = 3

. Therefore $x = (0, - 1, 1)^{T}$ . Substitution in the original system returns

(3, - 2, 0)^{T}

, as required.

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