Question

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

Answer 1

$A^n=\left[\begin{array}{ccc}1& n& \frac{n(n+1)}{2}\\ 0& 1& n\\ 0& 0& 1\end{array}\right]$

$M=A+A^2+A^3+.....+A^{20}$

$=\left[\begin{array}{ccc}\sum _{n=1}^{20}1& \sum _{n=1}^{20}n& \sum _{n=1}^{20}\frac{n(n+1)}{2}\\ 0& \sum _{n=1}^{20}1& \sum _{n=1}^{20}n\\ 0& 0& \sum _{n=1}^{20}1\end{array}\right]$
And
$\sum _{n=1}^{20}1=20,\sum _{n=1}^{20}n=\frac{20\times 21}{2}=210,\sum _{n=1}^{20}\frac{1}{2}n(n+1)=\frac{1}{2}[\sum _{n=1}^{20}{n}^2+\sum _{n=1}^{20}n]=\frac{1}{2}[\frac{20\times 21\times 41}{6}+210]=1540$

$M=\left[\begin{array}{ccc}20& 210& 1540\\ 0& 20& 210\\ 0& 0& 20\end{array}\right]$
Sum $=20+20+20+210+210+1540=2020$