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

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- Solution 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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