We consider the matrix $K \in \mathscr{M}_N(\mathbf{R})$ defined by $\forall (i,j) \in \llbracket 1;N \rrbracket^2, K[i,j] = p_{ij}$, and the random variable $Z_k$ representing the state of the system after $k$ impulses, with $Z_0$ being the certain variable with value 1.\\
Let $n \in \mathbf{N}$. Let $j \in \llbracket 1;N \rrbracket$, show that $P(Z_n = j) = K^n[1,j]$.\\
One may proceed by induction.