Interchanging two rows reverses the sign of an <alternating multilinear map>, so
$$
\boxed{\det C=-\det A}.
$$
In the <Leibniz formula for determinants>, the same result follows by composing every <permutation> with the corresponding <transposition>, which reverses its <sign of a permutation>[sign].

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / ia / Paper 1 / 2A / iii / Solution

An alternating multilinear map vanishes whenever two arguments are equal. Swapping two arguments reverses its sign.


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