A is an n times n invertible symmetric matrix prove A1 is al

A is an n times n, invertible, symmetric matrix, prove A^-1 is also symmetric.

As we know : I = I^T

So we can write : AA^-1 = (AA^-1)^T

AA^-1 = A^T(A^T)^-1

A^-1A = (A^T)^-1A^T ( As A^TB^T = B^TA^T)

we know A = A^T

So, substitute the above:

A^-1A = (A^T)^-1A

Now multiply both sides by A^-1:

A^-1AA^-1 = (A^T)^-1AA^-1

A^-1= (A^T)^-1

Ao, A^-1 is also symmteric