Let A=pmatrix1&0&01&1&01&1&1pmatrix and B=A^20.

Question

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

Accepted Answer

By induction, $A^n=\begin{pmatrix}1&0&0\n&1&0\\binom{n+1}2&n&1\end{pmatrix}$.

$A^{20}$'s first column is $\begin{pmatrix}1\20\\binom{21}2\end{pmatrix}=\begin{pmatrix}1\20\210\end{pmatrix}$.

Sum $=1+20+210=231$.

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