Problem regarding tensor

Question

Prove that the contraction of a tensor $A^l_m$ is a scalar or invariant.

My try:

Mixed tensor of rank 2 formula: $$\bar A^l_m=\frac{\partial \bar x^l}{\partial x^p}\frac{\partial x^q}{\partial \bar x^m}A^p_q$$

Then letting $l=m$ and also $p=q$

$$\bar A^l_l=\frac{\partial \bar x^l}{\partial x^p}\frac{\partial x^p}{\partial \bar x^l}A^p_p$$

Finally $\bar A^l_l=A^p_p$

Am I correct?

Answer 1

To be more precise, I'd write: $$ \bar{A}_\ell^\ell = \frac{\partial \bar{x}^\ell}{\partial x^p} \frac{\partial x^q}{\partial \bar{x}^\ell}A^p_q = \delta^q_pA^p_q = A_q^q $$ It holds in general for tensor contractions (see here).