Past exam of the year 1 of the computer science course of the University of Oxford / 2022 / Digital systems, linear algebra, introduction to formal proof resit / 6 / c / i / Solution

If $B^k=0$ with $k\ge 1$, put $S=I+B+\cdots +B^{k-1}$. The matrix geometric series telescopes on both sides:Thus $I-B$ is nonsingular, andThis is a finite polynomial identity for a nilpotent matrix; no convergence argument or bound $\Vert B\Vert <1$ is required.