Linear Algebra

Topic: Magic Squares 281(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 downthe 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)) islinear.(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 isalso a subspace of M n×n .6 [Beardon] Here is a slicker proof of the result of this Topic, when n 3. See theprior two exercises for some definitions and needed results.(a) First show that dim M n,0 = dim H n,0 + 2. To do this, consider the functionθ: M n → R 2 sending a matrix M to the ordered pair (Tr(M), Tr ∗ (M)). Specifically,consider the restriction of that map θ: H n → R 2 to the semimagic squares.Clearly its null space is M n,0 . Show that when n 3 this restriction θ is onto.(Hint: we need only find a basis for R 2 that is the image of two members of H n )(b) Let the function φ: M n×n → M (n−1)×(n−1) be the identity map except thatit drops the final row and column: φ(M) = ˆM where ˆm i,j = m i,j for alli, j ∈ {1 . . . n − 1}. The check that φ is linear is easy. Consider φ’s restriction tothe semimagic squares with magic number zero φ: H n,0 → M (n−1)×(n−1) . Showthat φ is one-to-one(c) Show that φ is onto.(d) Conclude that H n,0 has dimension (n − 1) 2 .(e) Conclude that dim(M n ) = n 2 − n