Let $\mathrm{A}=\left[{\mathrm{a}}_{\mathrm{ij}}\right]$ be a real matrix of order 3×3, such that ${\mathrm{a}}_{\mathrm{il}}+{\mathrm{a}}_{\mathrm{i}2}+{\mathrm{a}}_{\mathrm{i}3}=1$ for i = 1, 2, 3. Then, the sum of all the entries of the matrix A³ is equal to :

$\begin{array}{rcl}\mathrm{A}& =& \left[\begin{array}{ccc}{\mathrm{a}}_{11}& {\mathrm{a}}_{12}& {\mathrm{a}}_{13}\\ {\mathrm{a}}_{21}& {\mathrm{a}}_{22}& {\mathrm{a}}_{23}\\ {\mathrm{a}}_{31}& {\mathrm{a}}_{32}& {\mathrm{a}}_{33}\end{array}\right], \mathrm{X}=\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]\\ \mathrm{AX}& =& \left[\begin{array}{c}{\mathrm{a}}_{11}+{\mathrm{a}}_{12}+{\mathrm{a}}_{13}\\ {\mathrm{a}}_{21}+{\mathrm{a}}_{22}+{\mathrm{a}}_{23}\\ {\mathrm{a}}_{31}+{\mathrm{a}}_{32}+{\mathrm{a}}_{33}\end{array}\right]=\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]\\ \therefore \mathrm{AX}& =& \mathrm{X}\\ & & \mathrm{replace} \mathrm{X} \mathrm{by} \mathrm{AX}\\ {\mathrm{A}}^2\mathrm{X}& =& \mathrm{AX}=\mathrm{X}\\ & & \mathrm{replace} \mathrm{X} \mathrm{by} \mathrm{AX}\\ {\mathrm{A}}^3\mathrm{X}& =& \mathrm{AX}=\mathrm{X}\\ \mathrm{Let} {\mathrm{A}}^3& =& \left[\begin{array}{ccc}{\mathrm{x}}_1& {\mathrm{x}}_2& {\mathrm{x}}_3\\ {\mathrm{y}}_1& {\mathrm{y}}_2& {\mathrm{y}}_3\\ {\mathrm{z}}_1& {\mathrm{z}}_2& {\mathrm{z}}_3\end{array}\right]\\ \therefore {\mathrm{A}}^3\mathrm{X}& =& \mathrm{X}\\ \left[\begin{array}{c}{\mathrm{x}}_1+{\mathrm{x}}_2+{\mathrm{x}}_3\\ {\mathrm{y}}_1+{\mathrm{y}}_2+{\mathrm{y}}_3\\ {\mathrm{z}}_1+{\mathrm{z}}_2+{\mathrm{z}}_3\end{array}\right]& =& \left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]\\ \therefore \mathrm{Sum} \mathrm{of} \mathrm{all} \mathrm{the} \mathrm{elements} & =& 3\end{array}$