 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/b/solution

Use multiplicativity of the determinant and the determinant criterion for an invertible matrix:

0 \neq = det A = det L det U .

Neither factor can therefore be zero, so both matrices are nonsingular. The determinant of a triangular matrix is the product of its diagonal entries:

det L = \prod_{i = 1}^{N} L_{ii}, det U = \prod_{i = 1}^{N} U_{ii} .

Each product is nonzero, forcing every diagonal factor to be nonzero. These explicitly stated determinant properties justify both conclusions.

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