A matrix is said to be a semi-magic square if its row- sums and column sums (i.e., the sum of entries in an individual row or column) all add up to the same number. An example is

whose row and column sums are all equal to 15.
(a) Explain why the set of all semi-magic squares forms a subspace.
(b) Prove that the 3 × 3 permutation matrices (1.30) span the space of semi-magic squares. What is its dimension?
(c) A magic square also has the diagonal and anti- diagonal (running from top right to bottom left) also add up to the common row and column sum; the preceding 3×3 example is magic. Does the set of 3 × 3 magic squares form a vector space? If so, what is its dimension?
(d) Write down a formula for all 3 x 3 magic squares

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