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pow:problem5f25
Problem 5 (due Monday, November 3 )
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 Solution.
pow/problem5f25.txt · Last modified: 2025/11/08 13:07 by mazur