Part I 3. Let A be a (0,1)-matrix of order n satisfying the equati...

Question

Part I
3. Let A be a (0,1)-matrix of order n satisfying the equation A + Aᵀ = J - I. Prove that the term rank of A is at least n - 1.
7. Prove that the product of two doubly stochastic matrices is a doubly stochastic matrix.
8. Let A be a doubly stochastic matrix of order n. Let A' be a matrix of order n - 1 obtained by deleting the row and column of a positive element of A. Prove that per(A) > 0.

Part II
2. Show that the n eigenvalues of the matrix tI - aJ of order n are t with multiplicity n - 1 and t + an
4. Let A be a (0,1)-matrix of order n which satisfies the matrix equation AAᵀ = tI - aJ. Generalize the argument given in the text for a = 1 to prove that A is a normal matrix.
6. Verify that the incidence matrix A of the projective plane of order 2 satisfies per(A) = |det(A)| = 24.

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