Explanation Given Information: A collection of n × n matrices is said to be a convex set, if whenever A and B are in the set and t is a number between 0 to 1, then t A + ( 1 − t ) B is also in the set. Formula used: For a square matrix, if the sum of the entries in each column is one and also the sum of the entries in each row is one, then the matrix is said to be doubly stochastic matrix. Proof: Assume A and B are n × n doubly stochastic matrices, A = [ a 11 a 12 ⋯ a 1 n a 21 a 22 ⋯ a 2 n ⋮ ⋮ ⋮ ⋮ a n 1 a n 2 ⋯ a n n ] n × n And, B = [ b 11 b 12 ⋯ b 1 n b 21 b 22 ⋯ b 2 n ⋮ ⋮ ⋮ ⋮ b n 1 b n 2 ⋯ b n n ] n × n Therefore, ∑ a i j = 1 and ∑ b i j = 1 . Now, consider a matrix C = t A + ( 1 − t ) B . Where, 0 < t < 1 . Therefore, C = t A + ( 1 − t ) B = t [ a 11 a 12 ⋯ a 1 n a 21 a 22 ⋯ a 2 n ⋮ ⋮ ⋮ ⋮ a n 1 a n 2 ⋯ a n n ] + ( 1 − t ) [ b 11 b 12 ⋯ b 1 n b 21 b 22 ⋯ b 2 n ⋮ ⋮ ⋮ ⋮ b n 1 b n 2 ⋯ b n n ] = [ t a 11 t a 12 ⋯ t a 1 n t a 21 t a 22 ⋯ t a 2 n ⋮ ⋮ ⋮ ⋮ t a n 1 t a n 2 ⋯ t a n n ] + [ ( 1 − t ) b 11 ( 1 − t ) b 12 ⋯ ( 1 − t ) b 1 n ( 1 <

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To prove: The set of all n × n doubly stochastic matrices is a convex set.

Solution Summary: The author proves that the set of all ntimes n doubly stochastic matrices is a convex set.

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