 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2014/digital-systems-linear-algebra-introduction-to-formal-proof/6/c/solution

Solve two triangular linear systems, rather than forming an inverse matrix:

L y = b by forward substitution, Ux = y by back substitution .

y_{1} = b_{1}

y_{2} = b_{2} + 2 y_{1}

and

y_{3} = b_{3} - 3 y_{1} + y_{2}

x_{3} = - y_{3}, x_{2} = \frac{y _{2} - 2 x _{3}}{4}, x_{1} = y_{1} - 4 x_{2} - x_{3} .

All diagonal entries used as divisors are nonzero. The same LU decomposition can be reused for many right-hand vectors, with only these two triangular solves repeated.

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