We began Section 3.6 with the remark that a matrix A is invertible if and only if its determinant is nonzero: Now we want to show that this result also holds for matrices. This connection between determinants and invertibility is closely related to the fact that the determinant function “respects” matrix multiplication in the sense that
if A and B are matrices. Our first step is to show that Eq. (9) holds if A is an elementary matrix obtained from the identity matrix I by performing a single elementary row operation.
This is really just a restatement of Theorem 1. For instance, suppose that E is obtained from I by multiplying the pth row by c, so that Then Theorem 5 in Section 3.5 tells us that the product EB is the result of multiplying the pth row of B by c. Therefore,
and so we have verified Eq. (10) for the first of the three types of elementary matrices. The verifications for the other two types are similar.
Now, let A be an matrix whose invertibility we want to discuss, and let R be the reduced echelon form of A. If the elementary matrices correspond to the elementary row operations that reduce A to R, then
by Theorem 5 in Section 3.5. Recalling that every elementary matrix is invertible (Section 3.5), we can rewrite Eq. (11) as
where each is an elementary matrix. It now follows, by k applications of the lemma, that
This relation is the key both to the proof of (9) and to the proof of the following theorem.
If (as previously) R is the reduced echelon form of A, then Theorem 6 in Section 3.5 implies that
Because R is a square reduced echelon matrix, we see that either R is the identity matrix I and or R has an all-zero row and, consequently, Therefore,
Finally, because if E is an elementary matrix, it follows immediately from Eq. (13) that
Combining the statements in (14), (15), and (16), we see that A is invertible if and only if
So now we can add the statement det to the list of equivalent properties of nonsingular matrices stated in Theorem 7 of Section 3.5. Indeed, some texts define the square matrix A to be nonsingular if and only if det
If R is the reduced echelon form of A, then we see that
where and are elementary matrices. Hence
We now take the determinant of both sides, using the lemma stated earlier to “split off” the elementary matrices:
After steps, we get
The remainder of the proof depends on whether or not A is invertible.
If A is invertible, then so Eq. (12) yields
and also that In this case the meaning of Eq. (17) is precisely that
If A is not invertible, then by Theorem 2. Also, as we noted previously, the reduced echelon form R of A has an all-zero row in this case. Hence it follows from the definition of matrix multiplication that the product RB has an all-zero row and, therefore, that In this case Eq. (17) implies that Because both and the equation holds, and the proof is complete.
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