Is it hard to decide whether a matrix is a square of another matrix?

Question

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?

Answer 1

As for this question, is the following Karp redution correct?

Let the transfermation map $f:\mathbb Z_N\to M_2(\mathbb Z_N)$ be $f(m)=\left( m ~~~ 0\atop 0 ~~~ 1\right)$. Then, for a YES-instance $m=x^2\bmod N$, $$f(m)=\left( x ~~~ 0\atop 0 ~~~ 1\right)\left( x ~~~ 0\atop 0 ~~~ 1\right)\bmod N$$ is also a YES-instance naturally. For a NO-instance $m$, we need to prove $f(m)$ is a NO-instance too. Suppose $$f(m)=X^2\bmod N$$ be a YES-instance, then $$m=\det(f(m))=\det(X)^2\bmod N$$ becomes a YES-instance. This is a contradiction.