Let P= [ cc 3 2 & 1 2 -1 2 & 3 2 ], A= [ ll 1 & 1 0 & 1 ] and Q=P… - Vidyadip

MCQ

Let $\quad P=\left[\begin{array}{cc}\frac{\sqrt{3}}{2} & \frac{1}{2} \\ -\frac{1}{2} & \frac{\sqrt{3}}{2}\end{array}\right], A=\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right]$ and $Q=P Q P^{ T }$. If $P ^{ T } Q ^{2007} P =\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]$, then $2 a+b-3 c-4 d$ equal to $...................$.

A
$2007$
✓
$2005$
C
$2006$
D
$2004$

✓

Answer

Correct option: B.

$2005$

b
$Q = PAP ^{ T }$

$P ^{ T } \cdot Q ^{2007} \cdot P = P ^{ T } \cdot Q \cdot Q \ldots Q \cdot P$

$= P ^{ T }\left( PAP ^{ T }\right)\left( P \cdot AP ^{ T }\right) \ldots\left( PAP ^{ T }\right) P \cdot$

$\Rightarrow\left( P ^{ T } P \right) A \left( P ^{ T } P \right) A \ldots A \left( P ^{ T } P \right)$

$P ^{ T } \cdot P =\left[\begin{array}{cc}\sqrt{3} / 2 & -1 / 2 \\ 1 / 2 & \sqrt{3} / 2\end{array}\right]\left[\begin{array}{cc}-\sqrt{3} / 2 & 1 / 2 \\ -1 / 2 & \sqrt{3} / 2\end{array}\right]\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]= I$

$\therefore P ^{ T } \cdot Q ^{200 /} \cdot P = A ^{200 /}$

$A ^2=\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right]\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right]=\left[\begin{array}{ll}1 & 2 \\ 0 & 1\end{array}\right]$

$\therefore A ^{2007}=\left[\begin{array}{cc}1 & 2007 \\ 0 & 1\end{array}\right]=\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]$

$a =1, b =2007, c =0, d =1$

$2 a + b -3 c -4 d =2+2007-4=2005$

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