Let A = [a_{ij}] be a real matrix of order 3 | vedclass

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(C) Given $A = [a_{ij}]$ is a $3 	imes 3$ matrix such that the sum of elements in each row is $1$. Let $X = \begin{bmatrix} 1 \ 1 \ 1 \end{bmatrix}

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$3$

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(C) Given $A = [a_{ij}]$ is a $3 	imes 3$ matrix such that the sum of elements in each row is $1$. Let $X = \begin{bmatrix} 1 \ 1 \ 1 \end{bmatrix}$. Then $AX = \begin{bmatrix} a_{11} + a_{12} + a_{13} \ a_{21} + a_{22} + a_{23} \ a_{31} + a_{32} + a_{33} \end{bmatrix} = \begin{bmatrix} 1 \ 1 \ 1 \end{bmatrix} = X$. Thus,$AX = X$. Multiplying both sides by $A$,we get $A^2X = A(AX) = AX = X$. Multiplying by $A$ again,we get $A^3X = A(A^2X) = AX = X$. Let $A^3 = [b_{ij}]$. Then $A^3X = \begin{bmatrix} b_{11} + b_{12} + b_{13} \ b_{21} + b_{22} + b_{23} \ b_{31} + b_{32} + b_{33} \end{bmatrix} = \begin{bmatrix} 1 \ 1 \ 1 \end{bmatrix}$. The sum of all entries of $A^3$ is $(b_{11} + b_{12} + b_{13}) + (b_{21} + b_{22} + b_{23}) + (b_{31} + b_{32} + b_{33}) = 1 + 1 + 1 = 3$.

Let $A = [a_{ij}]$ be a real matrix of order $3 \times 3$,such that $a_{i1} + a_{i2} + a_{i3} = 1$,for $i = 1, 2, 3$. Then,the sum of all the entries of the matrix $A^3$ is equal to:

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