Linear Algebra, 2020a

312 Chapter Three. Maps Between Spaces
c n+j = 0 for j ∈ {2 ... n− 2}. If a one does not appear in that column in ⃗ρ 2n+2
then we have c 2 =−c 2n+1 . In either case c 2 = 0, and thus c 2n+1 = c 2n+2 = 0
and c n+1 = c 2n = 0.
If the next block of n-many columns is not the last then similarly conclude
from its first column that c 3 = c n+1 = 0.
Keep this up until we reach the last block of columns, those numbered
(n − 1)n + 1 through n 2 . Because c n+1 = ···= c 2n = 0 column n 2 gives that
c n =−c 2n+1 = 0.
Therefore the rank of the matrix is 2n + 1, as required.
The classic source on normal magic squares is [Ball & Coxeter]. More on the
Lo Shu square is at [Wikipedia, Lo Shu Square]. The proof given here began
with [Ward].
Exercises
1 Let M be a 3×3 magic square with magic number s.
(a) Prove that the sum of M’s entries is 3s.
(b) Prove that s = 3 · m 2,2 .
(c) Prove that m 2,2 is the average of the entries in its row, its column, and in
each diagonal.
(d) Prove that m 2,2 is the median of M’s entries.
2 Solve the system a + b = s, c + d = s, a + c = s, b + d = s, a + d = s, and b + c = s.
3 Show that dim M 2,0 = 0.
4 Let the trace function be Tr(M) =m 1,1 + ···+ m n,n . Define also the sum down
the other diagonal Tr ∗ (M) =m 1,n + ···+ m n,1 .
(a) Show that the two functions Tr, Tr ∗ : M n×n → R are linear.
(b) Show that the function θ: M n×n → R 2 given by θ(M) =(Tr(M), Tr ∗ (m)) is
linear.
(c) Generalize the prior item.
5 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 H n is a subspace of M n×n .
(b) Show that the set H n,0 of n×n semimagic squares with magic number 0 is
also a subspace of M n×n .