Question

Let $A=\left[a_{ij}\right]$ 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:

• A. $1$
• B. $3$
• C. $2$
• D. $9$

Answer 1

The correct option is B $3$

Let a matrix $B=\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]$
$\therefore A\cdot B=B$
Sum of all entries of $A^3$ is equal to the only element of $B^T.A^3.B$
$\therefore B^T.A^3.B=B^T.A^2.\left(AB\right)=B^T.A^2.B=B^T.B=[3{]}_{1\times 1}$