Analytical calculation of specific symmetric 6x6 matrix

Question

I would like to calculate the inverse of this specific, $6\times6$, symmetric matrix analytically.

$\begin{bmatrix} 1&0&A&-C&A&C\\ 0&1&-C&B&C&B\\ A&-C&1&0&D&0\\ -C&B&0&1&0&E\\ A&C&D&0&1&0\\ C&B&0&E&0&1 \end{bmatrix}$

I know how to do it numerically but am wondering if it is possible to do it analytically? I found expressions for matrices up to $3\times3$, but no larger. Is this because it is not possible?

Answer 1

Thanks to @mjw I managed to make some progress.

By using the fact that the inverse also has to be symmetric, I came up with an answer in 24 steps. The following operations should be done on both the rows and then the columns. Let $Ei$ denote the $i$th row or column.

1. $E3 = E3 - E5$
2. $E4 = E4 + E6$
3. $E6 = E6 - 1/2 E4$
4. $E5 = E5 + 1/2 E3$
5. $E1 = E1 - (2A/(1+D)) E5$
6. $E6 = E6 - (C/(1-(2A^2)/(1+D))) E1$
7. $E2 = E2 + (C/(1-D))E3$
8. $E4 = E4 - (2B/(1-2C^2/(1-D)))E2$
9. $E4 = E4 - 1/2(1-E/2-C^2/(1-2A^2/(1+D)))^{-1} E6$

By this time you have a matrix with only nonzero elements on the diagonal. Then divide all rows by the nonzero element of that row and you obtain the identity matrix.

While this has been a nice exercise in linear algebra, the resulting matrix is very complicated and I have not been able to simplify it. But I'll leave this up for posterity in case it is of use to any one.