JEE MAIN - Mathematics (2021 - 27th July Evening Shift - No. 22)
If $$A = \left[ {\matrix{
1 & 1 & 1 \cr
0 & 1 & 1 \cr
0 & 0 & 1 \cr
} } \right]$$ and M = A + A2 + A3 + ....... + A20, then the sum of all the elements of the matrix M is equal to _____________.
Answer
2020
Explanation
$${A^n} = \left[ {\matrix{
1 & n & {{{{n^2} + n} \over 2}} \cr
0 & 1 & n \cr
0 & 0 & 1 \cr
} } \right]$$
So, required sum
$$ = 20 \times 3 + 2 \times \left( {{{20 \times 21} \over 2}} \right) + \sum\limits_{r = 1}^{20} {\left( {{{{r^2} + r} \over 2}} \right)} $$
$$ = 60 + 420 + 105 + 35 \times 41 = 2020$$
So, required sum
$$ = 20 \times 3 + 2 \times \left( {{{20 \times 21} \over 2}} \right) + \sum\limits_{r = 1}^{20} {\left( {{{{r^2} + r} \over 2}} \right)} $$
$$ = 60 + 420 + 105 + 35 \times 41 = 2020$$
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