First slide

Transpose of matrix

Question

If $P = [\begin{matrix} \frac{\sqrt{3}}{2} & \frac{1}{2} \\ \frac{- 1}{2} & \frac{\sqrt{3}}{2} \end{matrix}], A = [\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}] a n d Q = P A P^{T}, t h e n P^{T} (Q^{2005}) P$ is equal to

Moderate

A

$[\begin{matrix} 1 & 2005 \\ 0 & 1 \end{matrix}]$
B

$[\begin{matrix} \sqrt{3} / 2 & 2005 \\ 1 & 0 \end{matrix}]$
C

$[\begin{matrix} 1 & 2005 \\ \sqrt{3} / 2 & 1 \end{matrix}]$
D

$[\begin{matrix} 1 & \sqrt{3} / 2 \\ 0 & 2005 \end{matrix}]$

Solution

We have,
Q = P A P^T
⇒ P^T Q = AP^T ( $∵$ as P^TP = I = PP^T)
$∴$ P^T Q²⁰⁰⁵ P = A P^T Q²⁰⁰⁴ P
= A²P^T Q²⁰⁰³ P = A³P^T Q²⁰⁰² P
= A²⁰⁰⁴P^T (QP)
= A²⁰⁰⁴P^T(PA)
(Q = P A P^T⇒ Q P = P A) = A²⁰⁰⁵
= $[\begin{matrix} 1 & 2005 \\ 0 & 1 \end{matrix}]$

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