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(C) Given $A = \begin{bmatrix} 1 & 0 & 0 \ 1 & 1 & 0 \ 1 & 1 & 1 \end{bmatrix}$.Calculating powers of $A$:$A^2 = \begin{bmatrix} 1 & 0 & 0 \ 2 & 1

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$231$

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(C) Given $A = \begin{bmatrix} 1 & 0 & 0 \ 1 & 1 & 0 \ 1 & 1 & 1 \end{bmatrix}$.Calculating powers of $A$:$A^2 = \begin{bmatrix} 1 & 0 & 0 \ 2 & 1 & 0 \ 3 & 2 & 1 \end{bmatrix}$$A^3 = \begin{bmatrix} 1 & 0 & 0 \ 3 & 1 & 0 \ 6 & 3 & 1 \end{bmatrix}$$A^4 = \begin{bmatrix} 1 & 0 & 0 \ 4 & 1 & 0 \ 10 & 4 & 1 \end{bmatrix}$By observing the pattern,for $A^n$,the first column elements are $1$,$n$,and $\frac{n(n+1)}{2}$.Thus,$A^n = \begin{bmatrix} 1 & 0 & 0 \ n & 1 & 0 \ \frac{n(n+1)}{2} & n & 1 \end{bmatrix}$.For $n = 20$:$A^{20} = \begin{bmatrix} 1 & 0 & 0 \ 20 & 1 & 0 \ \frac{20(21)}{2} & 20 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \ 20 & 1 & 0 \ 210 & 20 & 1 \end{bmatrix}$.The sum of the elements of the first column is $1 + 20 + 210 = 231$.

Let $A = \begin{bmatrix} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 1 & 1 \end{bmatrix}$ and $B = A^{20}$. Then the sum of the elements of the first column of $B$ is?

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