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Yes, the inverse of a symmetric matrix is also symmetric.
Take the symmetric matrix A, we have:
and
This gives:
Using the properties:
and 
We get:
Since
, we can perform the substitution to get:
Multiplying by
on both sides:
Proving that the inverse of a symmetric matrix is also symmetric.
Take the symmetric matrix A, we have:
and
This gives:
Using the properties:
We get:
Since
Multiplying by
Proving that the inverse of a symmetric matrix is also symmetric.