"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 _____________."

Question

Accepted Answer

"$${A^n} = \left[ {\matrix{ 1 & n & {{{{n^2} + n} \over 2}} \cr 0 & 1 & n \cr 0 & 0 & 1 \cr } } \right]$$

So, required sum

$$ = 20 \times 3 + 2 \times \left( {{{20 \times 21} \over 2}} \right) + \sum\limits_{r = 1}^{20} {\left( {{{{r^2} + r} \over 2}} \right)} $$

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

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 _____________.

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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 _____________.

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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 _____________.