A.1.3. Determinants

Source:    §7.3,  R.D’Inverno, “Introducing Einstein’s Relativity”, Clarendon (92)

Consider a matrix A={ a ij }  with determinant A=det| A | .

The Laplace expansion is:

        A δ i j = a ik A jk                                     (a)

where the cofactors A ij  is defined as

A ij = ( ) i+j det| α ij |                   (b)

where the minor matrix α ij  conjugate to element a ij  is obtained by striking out the ith row and jth column of A.

 

The inverse A 1 ={ a ij }  is given by

        a ij = 1 A A ji

so that

        A A 1 = a ik a kj = 1 A a ik A jk = δ i j =I

 

From (a), we have

        A a ij = A ij

so that

        A t = A a ij a ij t = A ij a ij t                            (c)

Now, from (a), we see that

        δ i k B i = a ij t A kj          [ no summation on i ]

where Bi is the matrix obtained from A by replacing all elements a ij  of the ith row with their derivatives a ij t .  Thus, (c) may also be written as

        A t = i B i                                                (d)           [cf. (A.10)]