Two $n$ by $n$ matrices $M,N$ with real entries are called strongly independent if the matrix $aI+bM+cN$ is invertible for any real numbers $a,b,c$ which are not all $0$. a) Show that if $MN=NM$ then $M,N$ are not strongly independent. b) For any $n$ which is a multiple of 4 construct two strongly independent matrices of size $n$. (Here $I$ denotes the identity matrix.) Three solutions were submitted: by Ashton Keith (Purdue University), Gerald Marchesi, and Alif Miah (a partial solution). For a detailed solution, a generalization of the problem and its connections to some deep results in topology see the following link {{:pow:2025fproblem5.pdf|Solution}}.