= Solution
Use multiplicativity of the <determinant> and the determinant criterion for an <invertible matrix>:
$$
0\ne\det A=\det L\det U.
$$
Neither factor can therefore be zero, so both matrices are nonsingular. The determinant of a <triangular matrix> is the product of its diagonal entries:
$$
\det L=\prod_{i=1}^N L_{ii},\qquad\det U=\prod_{i=1}^N U_{ii}.
$$
Each product is nonzero, forcing every diagonal factor to be nonzero. These explicitly stated determinant properties justify both conclusions.
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