Recall that the row operations (a)-(c) correspond to elementary matrices, say , , and (and recall how we constructed , , and in elementary matrices). Since , we obtain , , and . Then the properties of determinant over row operations can be summarized in

If is invertible, there is a series of elementary matrices so that . By applying the equation above repeatedly, we obtain , and therefore . Similarly if is not invertible, we obtain (why?). Together we conclude that

if and only if is invertible.

If is invertible, we can express (why?). Since 's are also elementary matrices, we obtain

Department of Mathematics