QJEEM21Ma8cca: JEE Main 2021 Mathematics - Class 12 JEEM21Ma8cca Solution: If A = [ 1 1 1 0 1 1 0 0 1 ] and M = A + A 2 + A 3 + ....... +…

Question

Verified Solution • 2025–26 Session moderate

If

A = [\begin{matrix} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{matrix}]

and M = A + A² + A³ + ....... + A²⁰, then the sum of all the elements of the matrix M is equal to _____________.

Accepted Answer

A^{n} = [\begin{matrix} 1 & n & \frac{n^{2} + n}{2} \\ 0 & 1 & n \\ 0 & 0 & 1 \end{matrix}]

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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