JEE Mains · Maths · STD 12 - 3 and 4 . metrices and determinant
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 \(2121\)
Answer & Solution
Correct Answer
(B) \(2020\)
Step-by-step Solution
Detailed explanation
\(A^{n}=\left[\begin{array}{lll}1 & n & \frac{n^{2}+n}{2} \\ 0 & 1 & n \\ 0 & 0 & 1\end{array}\right]\) So, required sum \(=20 \times 3+2 \times\left(\frac{20 \times 21}{2}\right)+\sum_{\mathrm{r}=1}^{20}\left(\frac{\mathrm{r}^{2}+\mathrm{r}}{2}\right)\)…
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