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 $M$ is equal to ______

JEE Main · 2021 · 27 Jul 2021 · Shift 2 · Matrices

A = 100110111

and

M = A + A^{2} + A^{3} + \dots . + A^{20}

, then the sum of all the elements of the matrix

M

is equal to ______

Answer: 2020

Matrix

A

is upper triangular with all diagonal entries equal to 1.

Compute powers:

A^{2} = 100210321

A^{3} = 100310631

.

In general,

A^{k} = 100 k 10 \frac{k ( k + 1 )}{2} k 1

M = \sum_{k = 1}^{20} A^{k} = 2000 \sum_{k = 1}^{20} k 200 \sum_{k = 1}^{20} \frac{k ( k + 1 )}{2} \sum_{k = 1}^{20} k 20

\sum_{k = 1}^{20} k = \frac{20 \cdot 21}{2} = 210

\sum_{k = 1}^{20} \frac{k ( k + 1 )}{2} = \frac{1}{2} \sum_{k = 1}^{20} (k^{2} + k) = \frac{1}{2} (\frac{20 \cdot 21 \cdot 41}{6} + 210) = 1540

.

Sum of all elements:

20 + 210 + 1540 + 0 + 20 + 210 + 0 + 0 + 20 = 2020

JEE Main · 2021 · 27 Jul 2021 · Shift 2 · Integer answer