Let X=pmatrix111pmatrix and A=pmatrix-1&2&30&1&60&0&-1pmatrix.

Question

Let $X=\begin{pmatrix}1\1\1\end{pmatrix}$ and $A=\begin{pmatrix}-1&2&3\0&1&6\0&0&-1\end{pmatrix}$. For $k\in\mathbb{N}$, if $X^TA^kX=33$, then $k$ is equal to \_\_\_\_\_\_\_\_.

Answer: 10

Accepted Answer

Direct computation shows $A^2=\begin{pmatrix}1&0&6\0&1&0\0&0&1\end{pmatrix}$, so $A^{2m}=\begin{pmatrix}1&0&6m\0&1&0\0&0&1\end{pmatrix}$.

For even $k=2m$: $X^TA^kX=\begin{pmatrix}1&1&1\end{pmatrix}\begin{pmatrix}1&0&6m\0&1&0\0&0&1\end{pmatrix}\begin{pmatrix}1\1\1\end{pmatrix}=(1+6m)+1+1=3+6m$.

Setting $3+6m=33\Rightarrow m=5\Rightarrow k=10$.

(Checking odd $k$ leads to non-integer $k$, so $k=10$ is the valid solution.)

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