# Denote the value of the determinant

1. Apr 9, 2012

### rbwang1225

In "A Short Course in General Relativity", I met a statement that says if we denote the value of the determinant $|g_{ab}|$ by $g$, then the cofactor of $g_{ab}$ in this determinant is $gg^{ab}$ and following this we can deduce ∂$_cg=$$($∂$_cg_{ab})gg^{ab}$.

First, I don't understand what the first argument is.
Second, if the first argument is true, I don't know how to derive the second one.

Any help would be appreciated!

Last edited: Apr 9, 2012
2. Apr 9, 2012

### Dickfore

Do you know what a cofactor of a matrix is?

Do you know how to relate the inverse matrix to the cofactors?

Given the Laplace expansion:
$$g = \sum_{a, b}{g_{a b} \, \mathrm{Cof}_{a, b}}$$
what element(s) of this sum contain the term $g_{i j}$ for fixed indices i and j?

Then, what would:
$$\frac{\partial g}{\partial g_{i j}} = ?$$
be?

Knowing these partial derivatives, w.r.t. every element of the metric tensor, can you write the total differential of the determinant g?

This is a very important step in studying GR and you should remember it.