Source: §7.3, R.D’Inverno, “Introducing Einstein’s Relativity”, Clarendon (92)
Consider a matrix with determinant .
where the cofactors is defined as
where the minor matrix conjugate to element is obtained by striking out the ith row and jth column of A.
The inverse is given by
From (a), we have
Now, from (a), we see that
[ no summation on i ]
where Bi is the matrix obtained from A by replacing all elements of the ith row with their derivatives . Thus, (c) may also be written as
(d) [cf. (A.10)]