A surprising matrix problem
March 14, 2024 4 minute read
In this short article I will discuss a cute mathematical problem that I discovered while reading “Polya’s Footsteps: Miscellaneous Mathematical Expositions” by the Canadian mathematician Ross Honsberger. If you’re not familiar with Honsberger’s work, he is a well-known author in the field of recreational mathematics. The one and only Edsger W. Dijkstra referred to Honsberger’s work as “delightful”.
The problem was asked during the First Round of the Spanish Math Olympiad in 1988, and if you own the book, you can find it at page 9.
Prove that the sum \(S\) builds up to the exact total no matter what entries (\(x_1, x_2, ...\)) might be taken. So the sum \(S\) is always the same.
I randomly selected \(3\), then removed its row and column. The matrix \(A\) becomes: \(\begin{pmatrix} 4 & 5 \\ 7 & 8 \end{pmatrix}\). Then I randomly selected \(7\), removed its row and column, so that \(A\) becomes \(A=\begin{pmatrix} 5 \end{pmatrix}\). On this run the sum \(S_A=3+7+5=15\).
