 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2017/digital-systems-linear-algebra-introduction-to-formal-proof/5/c/ii/solution

Introduce

z = Ux

. Solve

L z = b

Ux = z

by back substitution. For these factors, the recurrences are

z_{1} = b_{1}, z_{2} = b_{2} - z_{1}, z_{3} = b_{3} - 4 z_{2},

followed by

x_{3} = - z_{3} /4, x_{2} = z_{2} - 2 x_{3}, x_{1} = z_{1} - x_{2} - x_{3} .

The nonzero diagonal entries make both triangular solves well-defined. No matrix inverse is calculated.

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