Symmetric/Antisymmetric Co and Contravariant Tensors

Question

Show that if the contravariant tensor $A^{ab}$ is symmetric and the covariant tensor $B_{ab}$ is antisymmetric, then

$A^{ab}B_{ab}$ $= 0$

I have tried plugging in and expanding definitions

$A^{ab}$$=$$1\over{2}$$(A^{ab}+A^{ba})$

$B_{ab}$$=$$1\over{2}$$(B_{ab}-B_{ba})$

but I fail to equate to zero. I have also tried using in addition with above

$A^{ab}$$=$$A^{ba}$

$B_{ab}$$=$$-B_{ba}$

Any help is appreciated

Thanks

Answer 1

Please verify the following calculation. $$ A^{ab}B_{ab} = A^{ab}(-B_{ba}) = -A^{ab}B_{ba} = -A^{ba}B_{ba} = -A^{ab}B_{ab} $$ Each step is either a simple algebraic manipulation or uses the assumed properties of $A$ and $B$. This calculation implies what you want.