Question

Let $x \, \in \, R$ and let $P = \begin{bmatrix}1&1&1\\\ 0&2&2\\\ 0&0&3\end{bmatrix},\quad Q = \begin{bmatrix}2&x;&x;\\\ 0&4&0\\\ x&x;&6\end{bmatrix}$ and $R = PQP^{-1}$. Then which of the following options is/are correct?

• A. There exists a real number x such that PQ = QP
• B. detR = det $\begin{bmatrix}2&x;&x;\\\ 0&4&0\\\ x&x;&5\end{bmatrix} +8,$ for all $x \, \in \, R$
• C. For x = 0,if R $\begin{bmatrix}1\\\ a\\\ b\end{bmatrix} = 6 \begin{bmatrix}1\\\ a\\\ b\end{bmatrix},$ then a + b = 5
• D. For x = 1, there exists a unit vector $\alpha\hat{i} + \beta\hat{j} +\gamma\hat{k}$ for which $R\begin{bmatrix}\alpha\\\ \beta\\\ \gamma\end{bmatrix} = \begin{bmatrix}0\\\ 0\\\ 0\end{bmatrix}$

Answer 1

Answer: (C)

For x = 0,if R $\begin{bmatrix}1\\\ a\\\ b\end{bmatrix} = 6 \begin{bmatrix}1\\\ a\\\ b\end{bmatrix},$ then a + b = 5

Answer 2

det($R$) = det($PQP^{-1}$) = (det $P$)(det$Q$) $\left(\frac{1}{det\,P}\right)$ $=$ det $Q$ $=48-4x^{2}$ for $x = 1$ det $\left(R\right)-44 \ne 0$ $\therefore$ for equation $R\begin{bmatrix}\alpha\\\ \beta\\\ \gamma\end{bmatrix}=\begin{bmatrix}0\\\ 0\\\ 0\end{bmatrix}$ We will have trivial solution $\alpha=\beta=\gamma$ $PQ = QP$ $PQP^{-1} = Q$ $R = Q$ No value of x. det $\begin{bmatrix}2&x;&x;\\\ 0&4&0\\\ x&x;&5\end{bmatrix}+8$ $=\left(40-4x^{2}\right)+8=48-4x^{2}-det R\,\forall\,x\,\in\,R$ $R=\begin{bmatrix}2&1&2/3\\\ 0&4&4/3\\\ 0&0&6\end{bmatrix}$ $\left(R-6I\right)\begin{bmatrix}1\\\ a\\\ b\end{bmatrix}=O$ $\Rightarrow-4+a+\frac{2b}{3}=0$ $-2a+\frac{4b}{3}=0$ $\Rightarrow a=2\,b=3$ $a+b=5$