Question

Is the inverse of a symmetric matrix also symmetric?

Answer 1

Yes, the inverse of a symmetric matrix is also symmetric.

Take the symmetric matrix A, we have:

$AA^{-1} = I$

and

$I^{T} = I$

This gives:

$(AA^{-1})^{T} = AA^{-1}$

Using the properties:

$AA^{-1} = A^{-1}A$ and $(AA^{-1})^{T} = (A^{-1})^{T}A^{T}$

We get:

$(A^{-1})^{T}A^{T} = A^{-1}A$

Since $A^{T} = A$, we can perform the substitution to get:

$(A^{-1})^{T}A = A^{-1}A$

Multiplying by $A^{-1}$ on both sides:

$(A^{-1})^{T}AA^{-1} = A^{-1}AA^{-1}$

$(A^{-1})^{T}I = A^{-1}I$

$(A^{-1})^{T} = A^{-1}$

Proving that the inverse of a symmetric matrix is also symmetric.