I was practicing some old tensor calculus exams and had to answer this question (paraphrased, it has multiple subquestions):
If $\eta_{ij}$ is the component of the metric $H$ on a 2D Riemannian manifold (i.e. $H=\eta_{ij} dx^i \otimes dx^j$) and $w_i$ is the componont of a smooth covector field $\Omega=w_i dx^i$, construct a symmetric tensor $\Gamma=\gamma_{ij}dx^i \otimes dx^j=(\eta_{ij}-w_i w_j)dx^i \otimes dx^j$.
Show that $det (\gamma_{ij})$ is equal to $det(\eta_{ij}) (1-\eta^{ij}w_i w_j)$.
In the solutions the answer is just "A direct computation shows this". I have verified the correctness of the final formula by just bruteforcing the summations in 2D, but I want to be able to verify this in index notation. So far I've found two threads which contain (a variant) of this formula: Formula for determinant of sum of matrices and Expressing the determinant of a sum of two matrices?. However, one proves it using standard matrix notation and the other only proves it in the n-dimensional cases which I found a quite hard to follow since this is my first actual mathematics course (physicist). (The formula is, for an invertible matrix A, $det(A+B)=det(A)+det(B)+trace(A^{-1} B)$, this is just a special case of that formula)
So far I've got this:
$det(\gamma_{ij})=\frac{1}{2}\epsilon^{i_1 i_2}\epsilon^{j_1 j_2}\gamma_{i_1 j_1}\gamma_{i_2 j_2}=\frac{1}{2}\epsilon^{i_1 i_2}\epsilon^{j_1 j_2}(\eta_{{i_1 j_1}}-w_{i_1}w_{j_1})(\eta_{{i_2 j_2}}-w_{i_2}w_{j_2})=\frac{1}{2}\epsilon^{i_1 i_2}\epsilon^{j_1 j_2}(\eta_{{i_1 j_1}}\eta_{{i_2 j_2}}+w_{i_1}w_{j_1}w_{i_2}w_{j_2}-\eta_{{i_1 j_1}}w_{i_2}w_{j_2}-\eta_{{i_2 j_2}}w_{i_1}w_{j_1})=$
$=det(\eta_{ij})+det(w_i w_j)-\frac{1}{2}\epsilon^{i_1 i_2}\epsilon^{j_1 j_2}(\eta_{{i_1 j_1}}w_{i_2}w_{j_2}+\eta_{{i_2 j_2}}w_{i_1}w_{j_1})$
The determinant of the ww matrix is obviously zero, so now all that's left to show is
$\frac{1}{2}\epsilon^{i_1 i_2}\epsilon^{j_1 j_2}(\eta_{{i_1 j_1}}w_{i_2}w_{j_2}+\eta_{{i_2 j_2}}w_{i_1}w_{j_1})=det(\eta_{ab})\eta^{ij}w_i w_j=\frac{1}{2}\epsilon^{i_1 i_2}\epsilon^{j_1 j_2}\eta_{{i_1 j_1}}\eta_{{i_2 j_2}}\eta^{\alpha \beta}w_{\alpha}w_\beta$
Now, in my understanding the terms in the brackets on the l.h.s are equal to eachother after summing with the antisymmetric symbols, so all that's left to do is show, using only valid index juggling tricks, that $\eta_{{i_1 j_1}}w_{i_2}w_{j_2}$ can be written as $\frac{1}{2}\eta_{{i_1 j_1}}\eta_{{i_2 j_2}}\eta^{\alpha \beta}w_{\alpha}w_\beta$. This is where I'm stuck, however of course I could have made (multiple) mistakes before arriving there but I can't seem to find it and any help would be greatly appreciated. Edit: I got it now and it rests upon some property of the antisymmetric symbol; $\epsilon^{a b} c_{a i} c _{b k}= \epsilon_{i k} det (c) $ but I'm too lazy too type it and I'm not allowed to post pictures yet.
