Randomized Algorithm with matrices
Let's start by assuming that we have two bit vectors $A,B$ of size $m$ instead of two matrices – the matrix structure is unhelpful.

Let $w \in \{\pm1\}^m$ be chosen randomly, and let $\varphi = x \mapsto \langle x,w \rangle \pmod{3}$. If $A,B$ differ on the coordinates in $S$ then
$$ \varphi(A) - \varphi(B) = \sum_{i \in S} (A_i - B_i) w_i \pmod{3}. $$
The distribution of this is the same as the distribution of
$$ \Delta = \sum_{i \in S} w_i \pmod{3}. $$
Suppose $s = |S|$. In order to compute the distribution of $\Delta$, let $\omega$ be a primitive third root of unity, and consider
$$ \left(\frac{\omega + \omega^{-1}}{2}\right)^s = \frac{(-1)^s}{2^s} = \left(\frac{1}{3} + \frac{(-1)^s}{3 \cdot 2^{s-1}}\right) \omega^0 + \left(\frac{2}{3} - \frac{(-1)^s}{3 \cdot 2^{s-1}}\right) \frac{\omega + \omega^{-1}}{2}. $$
For concreteness, here is the probability that $\Delta = 0$ for $s = 0,1,2,\ldots$:
$$ 1,0,\frac{1}{2},\frac{1}{4},\ldots $$
These probabilities tend to $1/3$ exponentially fast, alternately being above and below it. In particular, if $s = 1$ then $\Pr[\Delta = 0] = 0$, and otherwise $\Pr[\Delta = 0] \geq 1/4$.

This suggests the following algorithm, in the case in which the two computers are allowed shared randomness:


Both computers extract $k$ random functionals $\varphi_1,\ldots,\varphi_k$ from the common randomness.
Computer 1 sends $\varphi_1(A),\ldots,\varphi_k(A)$ to Computer 2.
Computer 2 computes $\varphi_1(B),\ldots,\varphi_k(B)$, and outputs Yes if $\varphi_i(A) \neq \varphi_i(B)$ for all $1 \leq i \leq k$.


If the two vectors differ in exactly one location, then Computer 2 always outputs Yes. Otherwise, it outputs No with probability at least $1 - (3/4)^k$. Choosing $k = 3$, we get $1 - (3/4)^3 &gt; 1/2$.

What if the computers are not allowed shared randomness? A standard trick allows us to simulate shared randomness using an overhead of $O(\log m)$. In total, the communication complexity is $O(\log m)$.