Determinant function/Behavior under row operations/Fact/Proof

Proof

(1) and (2) follow directly from multilinearity.
(3) follows from fact.
To prove (4), we consider the situation where we add to the -th row the -multiple of the -th row, . Due to the parts already proven, we have


(5). If a diagonal element is , then set . We can add to the -th row suitable multiples of the -th rows, , in order to achieve that the new -th row is a zero row, without changing the value of the determinant function. Due to (2), this value is .

In case no diagonal element is , we may obtain, by several scalings, that all diagonal element are . By adding rows, we obtain furthermore the identity matrix. Therefore,