Yes, the inverse of a symmetric matrix is also symmetric.
Take the symmetric matrix A, we have:
[tex]AA^{-1} = I[/tex]
and
[tex]I^{T} = I[/tex]
This gives:
[tex](AA^{-1})^{T} = AA^{-1}[/tex]
Using the properties:
[tex]AA^{-1} = A^{-1}A[/tex] and [tex](AA^{-1})^{T} = (A^{-1})^{T}A^{T}[/tex]
We get:
[tex](A^{-1})^{T}A^{T} = A^{-1}A[/tex]
Since [tex]A^{T} = A[/tex], we can perform the substitution to get:
[tex](A^{-1})^{T}A = A^{-1}A[/tex]
Multiplying by [tex]A^{-1} [/tex] on both sides:
[tex](A^{-1})^{T}AA^{-1} = A^{-1}AA^{-1}[/tex]
[tex](A^{-1})^{T}I = A^{-1}I[/tex]
[tex](A^{-1})^{T} = A^{-1}[/tex]
Proving that the inverse of a symmetric matrix is also symmetric.
