Question

If $P=\begin{bmatrix}\frac{\sqrt{3}}{2}&\frac{1}{2}\\ -\frac{1}{2}&\frac{\sqrt{3}}{2}\end{bmatrix}, A=\begin{bmatrix}1&1\\ 0&1\end{bmatrix}$ and $Q=PAP^{T}$, then $P\left(Q^{2005}\right)P^{T}$ equal to

• A. $\begin{bmatrix}1&2005\\ 0&1\end{bmatrix}$
• B. $\begin{bmatrix}\sqrt{3}/2&2005\\ 1&0\end{bmatrix}$
• C. $\begin{bmatrix}1&2005\\ \sqrt{3}/2&1\end{bmatrix}$
• D. $\begin{bmatrix}1&\sqrt{3}/2\\ 0&2005\end{bmatrix}$

Answer 1

Answer: (A)

Given $Q = PAP^T$
$\Rightarrow P^{T}Q=AP^{T}, \left(\because PP^{T}=I\right)$
$\Rightarrow P^{T}Q^{2005}P=AP^{T}Q^{2004}P=AP^{T}Q^{2003}PA$
$\left(\because Q=PAP^{T} \Rightarrow QP=PA\right)$
$=AP^{T}Q^{2002}PA^{2}=AP^{T}PA^{2004}$
$=AIA^{2004}=A^{2005}=\begin{bmatrix}1&1\\ 0&1\end{bmatrix}^{^{2005}}=\begin{bmatrix}1&2005\\ 0&1\end{bmatrix}$