Math Challenge - February 2020

[archaic]

Oh man, thank you!

You are welcome!

 

[fishturtle1]

For #5, Let $R$ be a commutative ring. To say a subset $I \subseteq R$ is an ideal of $R$ means that $I$ is nonempty, $I$ is closed under addition and that for any $r \in R$ and $i \in I$, we have $ri \in I$. And to say $\operatorname{char} R \neq 2$ means $2$ is not the smallest positive integer $n$ such that $n\cdot 1 = 0$. But I'm not sure how to relate this to matrix multiplication. Would it be possible to get another hint, please?

 

[fresh_42]

For #5, Let $R$ be a commutative ring. To say a subset $I \subseteq R$ is an ideal of $R$ means that $I$ is nonempty, $I$ is closed under addition and that for any $r \in R$ and $i \in I$, we have $ri \in I$. And to say $\operatorname{char} R \neq 2$ means $2$ is not the smallest positive integer $n$ such that $n\cdot 1 = 0$. But I'm not sure how to relate this to matrix multiplication. Would it be possible to get another hint, please?

The matrix has a certain additive decomposition (*) of which one part is in a certain ideal. The decomposition series of that ideal together with a certain property of (*) then solves the question.