Question:
If $\mathrm{A}=\left[\begin{array}{lll}1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1\end{array}\right]$ and $\mathrm{M}=\mathrm{A}+\mathrm{A}^{2}+\mathrm{A}^{3}+\ldots .+\mathrm{A}^{20}$
then the sum of all the elements of the matrix $M$ is
equal to____________
Solution:
$A^{n}=\left[\begin{array}{ccc}1 & n & \frac{n^{2}+n}{2} \\ 0 & 1 & n \\ 0 & 0 & 1\end{array}\right]$
So, required sum
$=20 \times 3+2 \times\left(\frac{20 \times 21}{2}\right)+\sum_{\mathrm{r}=1}^{20}\left(\frac{\mathrm{r}^{2}+\mathrm{r}}{2}\right)$
$=60+420+105+35 \times 41=2020$
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