Let $M$ be a set of matrices of integers. Let $R$ be the relation on $M$ defined as follows: For any two $m_1, m_2 \in M, (m_1, m_2) \in R$ iff the matrix multiplication $m_1 \cdot m_2$ is defined.
Part (a) - Given any element $m_1 \in M$, suppose that $(m_1, m_1) \in R$. What does that tell you about the shape of $m_1$?
For this part I'm pretty confident that the shape is a square.
Part (b) - Suppose that the elements of $M$ are chosen such that $R$ is reflexive. Explain why it must be true that $R$ is an equivalence relation.
From part (a), you know all the elements of $M$ are square matrices. You just need to show that the relation is therefore symmetric and transitive.
Adding more detail:
We know all elements of $M$ are square matrices.
Symmetry: If $(m_1, m_2) \in R$, then the matrix multiplication $m_1 m_2$ is defined. Therefore both matrices must be the same size ($n \times n$ for some $n$). What can you conclude about $(m_2, m_1)$?
Transitivity: If $(m_1, m_2) \in R$, then by the same argument, $m_1$ and $m_2$ must be the same size ($n \times n$ for some $n$). Similarly if $(m_2, m_3) \in R$, then $m_2$ and $m_3$ are the same size. What can you say about $(m_1, m_3)$?