Let be an -by- matrix. Then is invertible if and only if .
27. Further properties of determinants
Recall that is invertible if and only if its determinant is nonzero. We'll prove the analogue of this for -by- matrices.
We'll also prove the following:
Proof of the invertibility criterion
Recall that is invertible if and only if its reduced echelon form (say ) is the identity matrix. To get from to , we used some row operations. The effect of these on the determinant is:
(type I) determinant unchanged,
(type II) determinant rescaled by a nonzero factor,
(type III) determinant switches sign.
Therefore differs from by a nonzero factor. In particular, if and only if .
If is invertible, , so .
If is not invertible, has a row of zeros ( is a square matrix in reduced echelon form, so if every row is nonzero then the leading entries have to go along the diagonal, and you get ; but we're assuming is not invertible). Since has a row of zeros, , so too.
Proof of multiplicativity of the determinant
We first prove in the case where is an elementary matrix.
If then is obtained from by , so (such a row operation doesn't change the determinant). We also have , so we can verify the formula in this case just by calculating both sides.
If then is obtained from by , so . We also have , so again we can verify the formula just by computing both sides.
This shows that whenever is an elementary matrix.
By induction, we can now show that whenever is a product of elementary matrices, in other words whenever is an invertible matrix.
If is not invertible, then (by the previous theorem) so to verify the formula in this case, we need to prove that . Suppose for a contradiction that . Then is invertible. But if is invertible then is an inverse for : This gives a contradiction, so we deduce that .
We have now checked the formula for all possible matrices , which proves the theorem.