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Question

Let A=[aij] be a real matrix of order 3×3, such that ai1+ai2+ai3=1, for i=1,2,3. Then, the sum of all the entries of the matrix A3 is equal to:

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Solution

Let a matrix B=111
AB=B
Sum of all entries of A3 is equal to the only element of BT.A3.B
BT.A3.B=BT.A2.(AB)=BT.A2.B=BT.B=[3]1×1

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