Linear Algebra

Topic: Magic Squares 279rows, columns, and diagonals of a matrix⎛ ⎞a b c⎜ ⎟⎝d e f⎠g h iadd to zero then we have an (2n + 2)×n 2 linear system.a + b + c = 0d + e + f = 0g + h + i = 0a + d + g = 0b + e + h = 0c + f + i = 0a + e + i = 0c + e + g = 0The matrix of coefficients for the particular cases of n = 3 and n = 4 arebelow, with the rows and columns numbered to help in reading the proof. Withrespect to the standard basis, each represents a linear map h: R n2 → R 2n+2 .The domain has dimension n 2 so if we show that the rank of the matrix is 2n + 1then we will have what we want, that the dimension of the null space M n,0 isn 2 − (2n + 1).1 2 3 4 5 6 7 8 9⃗ρ 1 1 1 1 0 0 0 0 0 0⃗ρ 2 0 0 0 1 1 1 0 0 0⃗ρ 3 0 0 0 0 0 0 1 1 1⃗ρ 4 1 0 0 1 0 0 1 0 0⃗ρ 5 0 1 0 0 1 0 0 1 0⃗ρ 6 0 0 1 0 0 1 0 0 1⃗ρ 7 1 0 0 0 1 0 0 0 1⃗ρ 8 0 0 1 0 1 0 1 0 01 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16⃗ρ 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0⃗ρ 2 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0⃗ρ 3 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0⃗ρ 4 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1⃗ρ 5 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0⃗ρ 6 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0⃗ρ 7 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0⃗ρ 8 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1⃗ρ 9 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1⃗ρ 10 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0