Let $T_{\mu\nu}$ be a rank $(0,2)$ tensor, $V^\mu$ a vector, and $U_\mu$ a covector. Using the definition of tensors based on the tensor transformation law, determine whether each of the following is a tensor. If it is, state its rank. $$ (a)\,\,A_{\mu\nu} \equiv \frac{\partial U_\mu}{\partial x^\nu},\quad(b)\,\,F_{\mu\nu} \equiv A_{[\mu\nu]},\text{where $A_{\mu\nu}$ is defined in (a)} $$
I have the tensor transformation law as: $$ T^{\prime i_1i_2\cdots i_p}_{\quad\quad\quad j_1j_2\cdots j_q} = \frac{\partial X^{\prime i_1}}{\partial X^{k_1}}\cdots\frac{\partial X^{\prime i_p}}{\partial X^{k_p}}\frac{\partial X^{m_1}}{\partial X^{\prime j_1}}\cdots\frac{\partial X^{m_q}}{\partial X^{\prime j_q}}T^{k_1k_2\cdots k_p}_{\quad\quad\quad m_1m_2\cdots m_q} $$
I am not sure how this would even look for a rank $(0,2)$ tensor. Would it be: $$ T^{\prime}_{\mu\nu} = \frac{\partial X^{m_1}}{\partial X^{\prime \mu}}\frac{\partial X^{m_2}}{\partial X^{\prime \nu}}T^{}_{m_1m_2} $$
I don't see how this would be useful in showing $(a)$ is a tensor.