If A, S ? Mn(F) with SA = In = AS show that for any B ? Fnx1 the system AX = B is consistent and has a unique solution. Let L = {A ? Mn(R) | all row sums of A are 1, that is Ai1 + Ai2 + ? ? ? + Ain = 1 for all i}. i) Show that A ? L ? [1, 1, . . . , 1]T ? Rnx1 is a solution of AX = X. ii) Use part i) to show: A, B ? L ? AB ? L. (This shows that if for all i, ?j Aij = ?j Bij = 1 then ?j (AB)ij = 1––not at all obvious!)
This question was answered on: May 23, 2022
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