5.11. The Product Formula for Determinants

The product of an m ´ n matrix A=( a ij )  and an n ´ p matrix B=( b ij )  is defined as the m ´ p matrix C=( c ij )  with components [see §4.15]

        c ij = k=1 n a ik b kj                             (5.13)

Lemma 5.10.

Given an m ´ n matrix A=( a ij )  and an n ´ p matrix B=( b ij ) , we have

        ( AB ) i = A i B                             (5.14)

which gives the ith row of AB as the product of the ith row of A with B.

Proof

Using Ai (Ai) to denote the ith row (column) of A, we can write eq(5.13) as the dot product

        c ij = A i B j

The ith row of C is therefore

        C i =( c i1 ,   c i2 ,,   c ip )   =( A i B 1   ,   A i B 2   ,  , A i B p )

                = A i ( B 1 , B 2 ,, B p )   = A i B                     QED.

Theorem 5.11.  Product Formula for Determinants.

For any two n ´ n matrices A and B, we have

        det( AB )=( detA )( detB )                        (5.15)

Proof

By Lemma 5.10, we have

        det( AB )=d[ ( AB ) 1 ,   ( AB ) 2 ,  , ( AB ) n ]   =d( A 1 B,   A 2 B,,   A n B )

For a fixed B, we define the function

        f( A )=det( AB )

In terms of the rows of A, this becomes

        f( A 1 ,, A n )=d( A 1 B,   A 2 B,,   A n B )

It is obvious that f satisfies Axioms 1 and 2 of a determinant function.  Furthermore,

        f( I )=d( B )

so that by the uniqueness Theorem 5.9, we have

f( A )=d( A )f( I )=d( A )d( B )                    QED.