International Mathematics Competition
for University Students
2023

IMC2023: Day 2, Problem 6

Problem 6. Ivan writes the matrix \(\displaystyle \begin{pmatrix} 2 & 3 \\ 2 & 4 \end{pmatrix}\) on the board. Then he performs the following operation on the matrix several times:

• he chooses a row or a column of the matrix, and
• he multiplies or divides the chosen row or column entry-wise by the other row or column, respectively.

Can Ivan end up with the matrix \(\displaystyle \begin{pmatrix} 2 & 4 \\ 2 & 3 \end{pmatrix}\) after finitely many steps?

Alex Avdiushenko, Neapolis University Paphos, Cyprus

Hint: Construct an invariant quantity that does not change during Ivan's prcedure.

  Solution