An n xn-matrix A is a said to be defiant if there is a fixed real...

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An n xn-matrix A is a said to be defiant if there is a fixed real number r such that the sum of each row, each column, the diagonal, and the anti-diagonal, are all equal to r. In this case, we will refer to r as the defiant sum of A. For example, the following matrices are all defiant: A₁ = 294 753 618 (r = 15) A₂ = [-1/3 5/6 0 1/6 -1/6 2/3 1/2 1/3-1/2 (r = 1/2) A3 = -1 0 11 20-2 -1 0 (r = 0) 1 For a fixed positive integer n ≥ 3, let D(n) denote the set of all n x n defiant matrices. Show that under the usual notion of matrix addition and scalar multiplication, D(n) is a vector space (and hence a subspace of M₁ (R)). An n xn-matrix A is a said to be defiant if there is a fixed real number r such that the sum of each row, each column, the diagonal, and the anti-diagonal, are all equal to r. In this case, we will refer to r as the defiant sum of A. For example, the following matrices are all defiant: A₁ = 294 753 618 (r = 15) A₂ = [-1/3 5/6 0 1/6 -1/6 2/3 1/2 1/3-1/2 (r = 1/2) A3 = -1 0 11 20-2 -1 0 (r = 0) 1 For a fixed positive integer n ≥ 3, let D(n) denote the set of all n x n defiant matrices. Show that under the usual notion of matrix addition and scalar multiplication, D(n) is a vector space (and hence a subspace of M₁ (R)).

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