Let $X=\left[\begin{array}{l}1 \ 1 \ 1\end{array} ight]$ and $A=\left[\begin{array}{ccc}-1 & 2 &

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Let $X=\left[\begin{array}{l}1 \ 1 \ 1\end{array}ight]$ and $A=\left[\begin{array}{ccc}-1 & 2 & 3 \ 0 & 1 & 6 \ 0 & 0 & -1\end{array}ight]$. For $k \in N$, if $X^{\prime} A^{k} X=[33]$, then $k$ is equal to

Professor PC Thomas Classes · Answer

(10) : We have given that,\ $X=\left[\begin{array}{l}1 \ 1 \ 1\end{array}ight], A=\left[\begin{array}{ccc}-1 & 2 & 3 \ 0 & 1 & 6 \ 0 & 0 & -1\end{array}ight]$\ $A^{2}=\left[\begin{array}{lll}1 & 0 & 6 \ 0 & 1 & 0 \ 0 & 0 & 1\end{array}ight], A^{3}=\left[\begin{array}{ccc}-1 & 2 & -3 \ 0 & 1 & 6 \ 0 & 0 & -1\end{array}ight]$\ $A^{4}=\left[\begin{array}{lll}1 & 0 & 12 \ 0 & 1 & 0 \ 0 & 0 & 1\end{array}ight], A^{5}=\left[\begin{array}{ccc}-1 & 2 & -9 \ 0 & 1 & 6 \ 0 & 0 & -1\end{array}ight]$\ $A^{6}=\left[\begin{array}{ccc}1 & 0 & 18 \ 0 & 1 & 0 \ 0 & 0 & 1\end{array}ight], A^{7}=\left[\begin{array}{ccc}-1 & 2 & -15 \ 0 & 1 & 6 \ 0 & 0 & -1\end{array}ight]$\ When $k$ is even, $A^{k}=\left[\begin{array}{ccc}1 & 0 & 3 k \ 0 & 1 & 0 \ 0 & 0 & 1\end{array}ight]$\ $X^{\prime} A^{k} X=33 \Rightarrow\left[\begin{array}{lll}1 & 1 & 1\end{array}ight]\left[\begin{array}{ccc}1 & 0 & 3 k \ 0 & 1 & 0 \ 0 & 0 & 1\end{array}ight]\left[\begin{array}{l}1 \ 1 \ 1\end{array}ight]=[33]$\ $\Rightarrow\left[\begin{array}{lll}1 & 1 & 1\end{array}ight]\left[\begin{array}{c}1+3 k \ 1 \ 1\end{array}ight]=[33]$\ $\Rightarrow 1+3 k+1+1=33 \Rightarrow k=10$\ When $k$ is odd.\ $X^{\prime} A^{k} X=[33] \Rightarrow X^{\prime} A \cdot A^{k-1} X=[33]$\ $\left[\begin{array}{lll}1 & 1 & 1\end{array}ight]\left[\begin{array}{ccc}-1 & 2 & 3 \ 0 & 1 & 6 \ 0 & 0 & -1\end{array}ight]\left[\begin{array}{ccc}1 & 0 & 3(k-1) \ 0 & 1 & 0 \ 0 & 0 & 1\end{array}ight]\left[\begin{array}{l}1 \ 1 \ 1\end{array}ight]=[33]$\ $\Rightarrow \quad-3 k+2+3+8=33 \Rightarrow k=\frac{-20}{3} 
otin N$\ Hence, $k=10$\