# Determinant function/Uniquely determined/Fact/Proof

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Proof

The
determinant
fulfills, due to
fact,
fact
and
fact,
all the given properties.

Uniqueness. For every matrix , there exists a sequence of elementary row operations such that, in the end, we get an upper triangular matrix. Hence, due to
fact,
the value of the determinant function is determined by the values on the upper triangular matrices. Therefore, after scaling and row addition, it is even determined by its value on the identity matrix.