Discussion
A square matrix fails to be invertible on one of these grounds. Which?
Answer: A
Correct answer: its determinant is zero
\( A^{-1} = \mathrm{adj}(A)/|A| \), so a zero determinant makes the inverse undefined; equivalently, the rows are dependent and the map \( A \) collapses space onto a lower dimension, which cannot be undone. A zero trace, skew-symmetry and individual zero entries are all compatible with invertibility - the identity matrix itself is full of zeros.
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