Question

If \( A = \begin{pmatrix} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{pmatrix} \) and \( M = A + A^2 + A^3 + \cdots + A^{20} \), then the sum of all the elements of the matrix \( M \) is equal to ________.

Hint:

For matrices of the form $I+B$ where B is nilpotent (i.e., $B^k=O$ for some k), the binomial theorem $(I+B)^n = I + nB + \frac{n(n-1)}{2}B^2 + .......$ becomes a finite sum and is a powerful tool for finding powers of the matrix.

Answer 1

Correct Answer: 2020

Key Observation:
$A=I+B$ where $B^3=O$ (nilpotent of index 3) \[ A^n=I+nB+\frac{n(n-1)}{2}B^2 \] Summing $A^1$ to $A^{20}$ element-wise gives \[ \sum M_{ij}=2020 \] \[ \boxed{2020} \]