Let $\mathrm{A}=\left[\mathrm{a}_{\mathrm{ij}}\right]$ be a real matrix of order $3 \times 3$, such that $\mathrm{a}_{\mathrm{il}}+\mathrm{a}_{\mathrm{i} 2}+\mathrm{a}_{\mathrm{i} 3}=1$, for $\mathrm{i}=1,2,3 .$ Then, the sum of all the entries of the matrix $\mathrm{A}^{3}$ is equal to :
Correct Option: , 3
$A=\left[\begin{array}{lll}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{array}\right]$
Let $x=\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]$
$A X=\left[\begin{array}{l}a_{11}+a_{12}+a_{13} \\ a_{21}+a_{22}+a_{23} \\ a_{31}+a_{32}+a_{33}\end{array}\right]=\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]$
$\Rightarrow \mathrm{AX}=\mathrm{X}$
Replace $\mathrm{X}$ by $\mathrm{AX}$
$\mathrm{A}^{2} \mathrm{X}=\mathrm{AX}=\mathrm{X}$
Replace $\mathrm{X}$ by $\mathrm{AX}$
$\mathrm{A}^{3} \mathrm{X}=\mathrm{AX}=\mathrm{X}$
$\operatorname{Let} A^{3}=\left[\begin{array}{lll}x_{1} & x_{2} & x_{3} \\ y_{1} & y_{2} & y_{3} \\ z_{1} & z_{2} & z_{3}\end{array}\right]$
$\mathrm{A}^{3}\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]=\left[\begin{array}{l}\mathrm{x}_{1}+\mathrm{x}_{2}+\mathrm{x}_{3} \\ \mathrm{y}_{1}+\mathrm{y}_{2}+\mathrm{y}_{3} \\ \mathrm{z}_{1}+\mathrm{z}_{2}+\mathrm{z}_{3}\end{array}\right]=\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]$
Sum of all the element $=3$