SUBIECTUL II Consider the matrices $A, B \in M_{n}(R)$ with the property that $B-A=I_{n}$. What condition must the matrix $\mathrm{A}$ satisfy so that, for any $k \in N^{*}$, the following equality holds:

$$
B+B^{2}+\ldots+B^{k}=k I_{n}+\frac{k(k+1)}{2} \cdot A ?
$$
Solution. For $k=1 \Rightarrow B=I_{n}+A$

$$
\begin{aligned}
\text { For } k= & 2 \Rightarrow B+B^{2}=2 I_{n}+3 A \\
& B+B^{2}=2 I_{n}+3 A+A^{2}, \text { so } A^{2}=O_{n} \Rightarrow A^{m}=O_{n}, \forall m \geq 2, m \in \mathbf{N} \\
\Rightarrow & B^{m}=\left(I_{n}+A\right)^{m}=I_{n}+m A \text { from which the relation. }
\end{aligned}
$$

## www.edums.ro