Creating Exponential Matrices

Question

Show thatI + (e^ t − 1)B$.

I'm not very sure how to even start on this question.

Answer 1

Since $B^2=B$, you have $B^n=B$, for each natural $n$. Therefore,\begin{align}\exp(tB)&=\operatorname{Id}+tB+\frac1{2!}t^2B^2+\frac1{3!}t^3B^3+\cdots\\&=\operatorname{Id}+tB+\frac1{2!}t^2B+\frac1{3!}t^3B+\cdots\\&=\operatorname{Id}+\bigl(t+\frac1{2!}t^2+\frac1{3!}t^3+\cdots\bigr)B.\end{align}Can you take it from here?