Question

If $A=\begin{bmatrix}1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1\end{bmatrix}$
$M=A+A^{2}+A^{3}+\ldots \ldots+A^{20}$, then the sum of all the elements of the matrix $M$ is equal to ____

Answer 1

$A^{n}=\begin{bmatrix}1 & n & \frac{n^{2}+n}{2} \\ 0 & 1 & n \\ 0 & 0 & 1\end{bmatrix}$
So, required sum
$=20 \times 3+2 \times\left(\frac{20 \times 21}{2}\right)+\displaystyle\sum_{r=1}^{20}\left(\frac{r^{2}+r}{2}\right) $
$=60+420+105+35 \times 41=2020$