Proving tensor symmetry under transformation

[Hendrick]

Homework Statement

Using indical notation, prove that a 2nd order symmetric tensor D remains symmetric when transformed into any other coordinate system.

Homework Equations

Tensor law of transformation (2nd order):
$$D'_{pq} = a_{pr}a_{qs}D_{rs}$$

The Attempt at a Solution

I think I'm required to prove that $$D'_{pq} = D'_{qp}$$ (where D is a symmetric 2nd order tensor)


$$D'_{pq} = a_{pr}a_{qs}D_{rs}$$
$$D'^{T}_{pq} = (a_{pr}a_{qs}D_{rs})^T$$
$$D'_{qp} = a_{qs}a_{pr}D_{sr}$$ (can someone please explain why when you transpose this, the a's swaps position but the D swaps indices?)

$$D_{rs} = D_{sr}$$ (as D is symmetric)

=>$$D'_{pq} = a_{pr}a_{qs}D_{rs} = a_{qs}a_{pr}D_{rs} = D'_{qp}$$

Thus $$D'_{pq} = D'_{qp}$$


Did I prove it correctly?

Thank you

 

[Hendrick]

Did I not write my question properly or does no one know?