Let A=[a_ij] be a real matrix of order 33, such that a_i1+a_i2+a_i3=1 for i=1,2,3.

Question

Let $A=[a_{ij}]$ be a real matrix of order $3	imes3$, 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

Accepted Answer

Let $X=\begin{pmatrix}1\1\1\end{pmatrix}$. The condition means $AX=X$. Then $A^3X=X$ as well (each row of $A^3$ sums to 1, since $A^2X=A(AX)=AX=X$ and similarly $A^3X=X$).

The sum of the entries of row $i$ of $A^3$ equals the $i$-th entry of $A^3X=X$, i.e. $1$. Summing over the 3 rows: total sum of all entries of $A^3$ is $1+1+1=3$.

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