Question

Let a matrix A = $\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$, P = $\begin{bmatrix} \frac{\sqrt{3}}{2} & \frac{1}{2} \

• \frac{1}{2} & \frac{\sqrt{3}}{2} \end{bmatrix}$, Q = PAP^T where P^T is transpose of matrix P. then P^T Q²⁰⁰⁵ P is-

• A. $\begin{bmatrix} 1 & 2005 \\ 0 & 1 \end{bmatrix}$
• B. $\frac{1}{4}$ $\begin{bmatrix} 1 + 2005\sqrt{3} & 6015 \\ 2005 & 1 - 2005\sqrt{3} \end{bmatrix}$
• C. $\frac{1}{4}$ $\begin{bmatrix} 1 + 2005\sqrt{3} & 2005 \\ 2005 & 1 - 2005\sqrt{3} \end{bmatrix}$
• D. $\begin{bmatrix} 2005 & 2005 \\ 0 & 1 \end{bmatrix}$

Answer 1

Answer: (A)

$\begin{bmatrix} 1 & 2005 \\ 0 & 1 \end{bmatrix}$

Answer 2

P^T Q²⁰⁰⁵ P = P^T (PAP^T)²⁰⁰⁵P

= P^T $\frac{\left\{ (PAP^{T})(PAP^{T})........(PAP^{T}) \right\}}{2005times}$P

= $\frac{(P^{T}P)A(P^{T}P)A(P^{T}P)........(P^{T}P)A(P^{T}P)}{2005times}$= A²⁰⁰⁵

A² =$\begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix}$, A³ = A²A = $\begin{bmatrix} 1 & 3 \\ 0 & 1 \end{bmatrix}$…. and so on.

A²⁰⁰⁵ = $\begin{bmatrix} 1 & 2005 \\ 0 & 1 \end{bmatrix}$Ž P^T Q²⁰⁰⁵ P = $\begin{bmatrix} 1 & 2005 \\ 0 & 1 \end{bmatrix}$