(A) Given $Q = PAP^T$. Note that $P$ is an orthogonal matrix,so $PP^T = I$ and $P^T = P^{-1}$.We want to compute $X = P^T Q^{2005} P$.Since $Q = PAP^T

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

(A) Given $Q = PAP^T$. Note that $P$ is an orthogonal matrix,so $PP^T = I$ and $P^T = P^{-1}$.We want to compute $X = P^T Q^{2005} P$.Since $Q = PAP^T$,we have $Q^n = (PAP^T)(PAP^T)...(PAP^T) = PA^n P^T$.Substituting this into the expression:$X = P^T (PA^{2005}P^T) P$$X = (P^T P) A^{2005} (P^T P)$Since $P^T P = I$,we get $X = I A^{2005} I = A^{2005}$.Given $A = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$,we use the property of this specific matrix: $A^n = \begin{bmatrix} 1 & n \\ 0 & 1 \end{bmatrix}$.Thus,$A^{2005} = \begin{bmatrix} 1 & 2005 \\ 0 & 1 \end{bmatrix}$.Therefore,the correct option is $A$.

Class 12 Mathematics 3 and 4 .Determinants and Matrices Mix Examples-Determinants and Matrices

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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 $P^T(Q^{2005})P$ is equal to

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JEE MAIN 2016 All JEE MAIN

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

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3 and 4 .Determinants and Matrices

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Mix Examples-Determinants and Matrices

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Let $m$ and $M$ be respectively the minimum and maximum values of $\left|\begin{array}{ccc}\cos ^{2} x & 1+\sin ^{2} x & \sin 2 x \\ 1+\cos ^{2} x & \sin ^{2} x & \sin 2 x \\ \cos ^{2} x & \sin ^{2} x & 1+\sin 2 x\end{array}\right|$. Then the ordered pair $(m, M)$ is equal to

DifficultJEE MAIN 2020

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Let $A=\left[\begin{array}{ll}2 & 3 \\ a & 0\end{array}\right], a \in R$ be written as $P+Q$ where $P$ is a symmetric matrix and $Q$ is a skew-symmetric matrix. If $\operatorname{det}(Q)=9$,then the modulus of the sum of all possible values of the determinant of $P$ is equal to:

MediumJEE MAIN 2021

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Let $A = \begin{bmatrix} 2 & -1 & -1 \\ 1 & 0 & -1 \\ 1 & -1 & 0 \end{bmatrix}$ and $B = A - I$. If $\omega = \frac{\sqrt{3}i - 1}{2}$, then the number of elements in the set $\{n \in \{1, 2, \ldots, 100\} : A^n + (\omega B)^n = A + B\}$ is equal to $..........$

DifficultJEE MAIN 2022

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Matrix $A$ is such that ${A^2} = 2A - I$,where $I$ is the identity matrix. Then for $n \ge 2$,${A^n} = $

Medium

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Consider the following relation $R$ on the set of real square matrices of order $3$. $R = \{(A,B) | A = P^{-1}BP \text{ for some invertible matrix } P\}$.
\textbf{Statement-$1$:} $R$ is an equivalence relation.
\textbf{Statement-$2$:} For any two invertible $3 \times 3$ matrices $M$ and $N$,$(MN)^{-1} = N^{-1}M^{-1}$.

MediumAIEEE 2011

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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 $P^T(Q^{2005})P$ is equal to

Similar Questions

Let $m$ and $M$ be respectively the minimum and maximum values of $\left|\begin{array}{ccc}\cos ^{2} x & 1+\sin ^{2} x & \sin 2 x \\ 1+\cos ^{2} x & \sin ^{2} x & \sin 2 x \\ \cos ^{2} x & \sin ^{2} x & 1+\sin 2 x\end{array}\right|$. Then the ordered pair $(m, M)$ is equal to

Let $A=\left[\begin{array}{ll}2 & 3 \\ a & 0\end{array}\right], a \in R$ be written as $P+Q$ where $P$ is a symmetric matrix and $Q$ is a skew-symmetric matrix. If $\operatorname{det}(Q)=9$,then the modulus of the sum of all possible values of the determinant of $P$ is equal to:

Let $A = \begin{bmatrix} 2 & -1 & -1 \\ 1 & 0 & -1 \\ 1 & -1 & 0 \end{bmatrix}$ and $B = A - I$. If $\omega = \frac{\sqrt{3}i - 1}{2}$, then the number of elements in the set $\{n \in \{1, 2, \ldots, 100\} : A^n + (\omega B)^n = A + B\}$ is equal to $..........$

Matrix $A$ is such that ${A^2} = 2A - I$,where $I$ is the identity matrix. Then for $n \ge 2$,${A^n} = $

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