Magic Squares

Let $M$ be a $3\!\times\!3$ magic square with magic number $s$.

• Prove that the sum of $M$’s entries is $3s$.

• Prove that $s=3\cdot m_{2,2}$.

• Prove that $m_{2,2}$ is the average of the entries in its row, its column, and in each diagonal.

• Prove that $m_{2,2}$ is the median of $M$’s entries.

Solve the system $a+b=s$, $c+d=s$, $a+c=s$, $b+d=s$, $a+d=s$, and $b+c=s$.

Show that $\dim\mathscr{M}_{2,0}=0$.

Let the trace function be $\operatorname{Tr}(M)=m_{1,1}+\cdots+m_{n,n}$. Define also the sum down the other diagonal $\operatorname{Tr}^{*}(M)=m_{1,n}+\cdots+m_{n,1}$.

• Show that the two functions $\operatorname{Tr},\operatorname{Tr}^{*}\colon\mathcal{M}_{n\!\times\!n}\to\mathbb{R}$ are linear.

• Show that the function $\theta\colon\mathcal{M}_{n\!\times\!n}\to\mathbb{R}^{2}$ given by $\theta(M)=(\operatorname{Tr}(M),\operatorname{Tr}^{*}(m))$ is linear.

• Generalize the prior item.

A square matrix is semimagic if the rows and columns add to the same value, that is, if we drop the condition on the diagonals.

• (a)  Show that the set of semimagic squares $\mathscr{H}_{n}$ is a subspace of $\mathcal{M}_{n\!\times\!n}$.     (b)  Show that the set $\mathscr{H}_{n,0}$ of $n\!\times\!n$ semimagic squares with magic number $0$ is also a subspace of $\mathcal{M}_{n\!\times\!n}$.