Question

If $A=\left[\begin{array}{ccc}1& 1& 1\\ 0& 1& 1\\ 0& 0& 1\end{array}\right]$
$M=A+A^2+A^3+\dots \dots +A^{20}$, then the sum of all the elements of the matrix $M$ is equal to ____

Answer 1

Answer: 2020

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