Throughout this whole section, let A, B denote real or complex matrices, u, v denote r-vectors, U, V denote matrices, denote the unit matrix of order r, and h denote an arbitrary real or complex number.

The Sherman-Morrison formula applies to inverting the modification of a given matrix with a dyad. If not a single dyad, but a finite sum of dyads (i.e. a matrix-product ) is added to A, then the Sherman-Morrison-Woodbury formula is applicable as a more general matrix equality. Further, it is not very difficult to give similar formulas for determinants rather than inverses, which we shall call „the Sherman-Morrison formula for the determinant", and the „Sherman-Morrison-Woodbury formula for the determinant".

(3.1)           .

We shall prove a more general formula that will occur in the next statement.

(3.2)           .

                   
                   
                   
                    .

(3.3)           ,

                    .

We omit the proof, which is elementary and is known from the determinant calculus.

(3.4)          .

Proof. Let us consider the matrices introduced in the previous statement. If then contains at least two columns selected from B, and therefore the assumption about the rank of B implies that

                    .

If and for , then

according to the determinant expansion theorem. Now we can write

                    .

(3.5)           .

Proof. The dyad is an matrix with rank equal to 1. According to Statement 3.4, we have

                    .

(3.6)           .

The proof is obvious, if we take into account that .

Corollary 3.7. If A and are invertible, then

                    .

The proof is evident from (3.6).

For the more general case, where the vectors u and v are replaced by the matrices U and V, statements, parallel to Corollaries 3.6-3.7 will also be formulated. Before doing this, we state the following lemma, which will help us to prove the generalization of Corollary 3.6.

Lemma 3.8. Let denote a real or complex matrix, and denote

(3.7)           .

Proof. Expand the determinant according to its last row, and then expand the given smaller determinants - with the exception of - according to their last columns. Then we obtain the sum standing on the right hand side of (3.7).

(3.8)          ,

(3.9)           .

Proof. We prove the theorem by induction on k. If k = 1, then the assertion reduces to the statement of Corollary 3.6.

Consider the modified matrix in the form

where and are r-vectors, and assume that the assertion of the theorem is true for k = 1, i.e.

                    .

We suppose temporarily, that is invertible. Then, according to Corollary 3.6,

                   
                   
                    .

According to Statement 3.2 we can obtain, that

where

and similarly

                    .

Thus

                   
                   
                   
                    .

Now we can write

         
                    .

Finally, by applying Lemma 3.8 for , we obtain that (3.8) is valid, provided that is invertible. As clearly , we can conclude that (3.8) is valid, if h is in a neighbourhood of zero. Since the expression

is a polynomial, i.e. analytical function of h, thus if it vanishes in a neighbourhood of zero, then it does so for any complex h. By this the proof is complete.

Corollary 3.10. If both A and are invertible, then is invertible as well.

Remark 3.11. The assertions of Statement 3.2 and Corollary 3.10 can be united in the following way: Suppose that A is invertible. In this case is invertible if and only if is invertible, and then

                    .

The Sherman-Morrison-Woodbury formula is cited in this form in [11], where is assumed. As we have seen, the restriction is not necessary to suppose.