If $B, C$ are square matrices of order $n$ and if $A=B+C, BC=CB, C^2=0$, then for any positive integer $p$, $A^{p+1}$.

$$\begin{gathered} \therefore A=B+C\Rightarrow A^{p+1}={(B+C)}^{p+1} \\ ={}^{p+1}C_0{B}^{p+1}+{}^{p+1}C_1{B}^{p}C+{}^{p+1}C_2{B}^{p-1}{C}^2+...+{}^{p+1}C_{p+1}{C}^{p+1} \\ =B^{p+1}+{}^{p+1}C_1{B}^{p}C+0+0+... [\because C^2=0\Rightarrow C^2=C^3=...=0] \\ =B^p[B+(p+1\left)C\right] \end{gathered}$$

Hence, $A^{p+1}=B^p[B+(p+1\left)C\right]$