Determinant/Zero, linear dependent and rank property/Fact/Proof
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Proof
The relation between rank, invertibility and linear independence was proven in fact. Suppose now that the rows are linearly dependent. After exchanging rows, we may assume that . Then, due to fact and fact, we get
Now suppose that the rows are linearly independent. Then, by exchanging of rows, scaling and addition of a row to another row, we can transform the matrix successively into the identity matrix. During these manipulations, the determinant is multiplied with some factor . Since the determinant of the identity matrix is , the determinant of the initial matrix is .