For a function $f(x)$, let $\left.\frac{d^x(f(x))}{d x^i}\right|_{x=a}$ denotes its $i$-th derivative with respect to $x$ evaluated at $x=a$. Consider the matrix $A=\left(a_{i, j}\right)_{n \times m}$ of $n( \gt 1)$ rows and $m( \gt 1)$ columns,
with
$a_{i, j}=\left(\left.\frac{d^i x^n}{d x^i}\right|_{x-1}\right) /\left(\left.\frac{d^j y^m}{d y^j}\right|_{y-1}\right)$
Let $p_i=\max \left\{a_{i, j}: j=1,2, \ldots, m\right\}, q_j=\min \left\{a_{i, j}: i=1,2, \ldots, n\right\}$, $p=\min \left\{p_i: i=1,2, \ldots, n\right\}$ and $q=\max \left\{q_j: j=1,2, \ldots, m\right\}$. Then

• A

  $p \gt q$
• Correct answer
  B

  $a_{1,1} \lt a_{2,2} \lt \cdots \lt a_{m, m}$, if $n \gt m$.
• Correct answer
  C

  $a_{n, m}=a_{n-1, m-1}=\cdots=a_{n-\ell+1, m-\ell+1}$, where $\ell=\min \{n, m\}$
• D

  $A$ is a symmetric matrix, if $m=n$.

Hence, the correct answer is options: 'B', 'C'.