If A=left[begin{array}{lll}1 & 1 & 1 0 & 1 & 1 0 & 0 & 1end{array}right] and

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3 and 4 .Determinants and Matrices

hard

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

A

$1010$

B

$2020$

C

$1414$

D

$2121$

(JEE MAIN-2021)

Solution

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

So, required sum

$=20 \times 3+2 \times\left(\frac{20 \times 21}{2}\right)+\sum_{\mathrm{r}=1}^{20}\left(\frac{\mathrm{r}^{2}+\mathrm{r}}{2}\right)$

$=60+420+105+35 \times 41=2020$

Standard 12

Mathematics

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