According to the well-know quadratic residue (QR) theory over integers, we know that it is hard to decide whether a given integer $m\in\mathbb Z_N$ is a quadratic residue (i.e., a square of another integer $x\in\mathbb Z_N$), without knowing the factorization of $N$.
Now, my question is: Without knowing the factorization of $N$, is it hard to decide whether a given matrix $M\in M_d(\mathbb Z_N)$ is a square of another matrix $X\in M_d(\mathbb Z_N)$ (i.e. $M=X^2\bmod N$) ?
In other words, when $d=1$, the so-called quadratic residue problem is a special case of my question. We know that it is hard when $d=1$, my question is: Is it hard for $d>1$?
Besides, is my following answer correct?