This paper quantifies the asymptotic order of the largest singular value of a centered random matrix built from the path of a Block Markov Chain (BMC). In a BMC there are $n$ labeled states, each state is associated to one of $K$ clusters, and the probability of a jump depends only on the clusters of the origin and destination. Given a path $X_0, X_1, \ldots, X_{T_n}$ started from equilibrium, we construct a random matrix $\hat{N}$ that records the number of transitions between each pair of states. We prove that if $ω(n) = T_n = o(n^2)$, then $\| \hat{N} - \mathbb{E}[\hat{N}] \| = Ω_{\mathbb{P}}(\sqrt{T_n/n})$. We also prove that if $T_n = Ω(n \ln{n})$, then $\| \hat{N} - \mathbb{E}[\hat{N}] \| = O_{\mathbb{P}}(\sqrt{T_n/n})$ as $n \to \infty$; and if $T_n = ω(n)$, a sparser regime, then $\| \hat{N}_Γ- \mathbb{E}[\hat{N}] \| = O_{\mathbb{P}}(\sqrt{T_n/n})$. Here, $\hat{N}_Γ$ is a regularization that zeroes out entries corresponding to jumps to and from most-often visited states. Together this establishes that the order is $Θ_{\mathbb{P}}(\sqrt{T_n/n})$ for BMCs.