If P = [ * 20 c 3 2 & 1 2 - 1 2 & 3 2 ], ,A = [ * 20 c 1&1 0&1 ] … - Vidyadip

MCQ

If $P = \left[ {\begin{array}{*{20}{c}}{\frac{{\sqrt 3 }}{2}}&{\frac{1}{2}}\\{ - \frac{1}{2}}&{\frac{{\sqrt 3 }}{2}}\end{array}} \right],\,A = \left[ {\begin{array}{*{20}{c}}1&1\\0&1\end{array}} \right]$ and $Q = PA{P^T}$, then ${P^T}({Q^{2005}})P$ equal to

✓
$\left[ {\begin{array}{*{20}{c}}1&{2005}\\0&1\end{array}} \right]$
B
$\left[ {\begin{array}{*{20}{c}}{\sqrt 3 /2}&{2005}\\1&0\end{array}} \right]$
C
$\left[ {\begin{array}{*{20}{c}}1&{2005}\\{\sqrt 3 /2}&1\end{array}} \right]$
D
$\left[ {\begin{array}{*{20}{c}}1&{\sqrt 3 /2}\\0&{2005}\end{array}} \right]$

✓

Answer

Correct option: A.

$\left[ {\begin{array}{*{20}{c}}1&{2005}\\0&1\end{array}} \right]$

a
(a) If $Q = PA{P^T}$

${P^T}Q = A{P^T}$, $({\rm{as}}\,\,P{P^T} = I)$

${P^T}{Q^{2005}}P = A{P^T}{Q^{2004}}P$

$ = {A^2}{P^T}{Q^{2003}}P$ $ = {A^3}{P^T}{Q^{2002}}P$ $ = {A^{2004}}{P^T}(QP)$

$ = {A^{2004}}{P^T}(PA)$ $(Q = PA{P^T} \Rightarrow QP = PA)$ $ = {A^{2005}}$

==> ${A^{2005}} = \left[ {\begin{array}{*{20}{c}}1&{2005}\\0&1\end{array}} \right]$.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "M.C.Q (1 Marks)" from STD 12 - 3 and 4 . determinant and metrices→STD 12 - 3 and 4 . determinant and metrices — all question types→Mathematics STD 12 Science question papers→CBSE Board STD 12 Science question papers→CBSE Board question paper generator→

Similar questions

A bag contains 5 brown and 4 white socks. A man pulls out two socks. The probability that these are of the sane colour is.

The number of real solution of the equation
$\sqrt{1+\cos2\text{x}}=\sqrt2\sin^{-1}(\sin\text{x}),-\pi\leq\text{x}\leq\pi$ is:

Let $f(x)$ be a positive function such that the area bounded by $y=f(x), y=0$ from $x=0$ to $x=a>0$ is $\mathrm{e}^{-\mathrm{a}}+4 \mathrm{a}^2+\mathrm{a}-1$. Then the differential equation, whose general solution is $y=c_1 f(x)+c_2$, where $c_1$ and $c_2$ are arbitrary constants, is :

The feasible region for an $\text{LPP}$ is shown below:

Let $Z = 3x - 4y$ be the objective function. Minimum of $Z$ occurs at

Maximize $Z = 7x + 11y,$ subject to $3\text{x}+5\text{y}\leq26,5\text{x}+3\text{y}\leq30,\text{x}\geq0,\text{y}\geq0.$

If $A = \left[ {\begin{array}{*{20}{c}}3&1\\{ - 1}&2\end{array}} \right]$, then ${A^2} = $

If ${\sin ^{ - 1}}x + {\sin ^{ - 1}}y + {\sin ^{ - 1}}z = \frac{{3\pi }}{2}$, then the value of ${x^{100}} + {y^{100}} + {z^{100}} - \frac{9}{{{x^{101}} + {y^{101}} + {z^{101}}}}$ is equal to

If $\text{V}=\frac{4}{3}\pi\text{r}^3,$ at What rate in cubic units is $V$ increasing when $\text{r}=10\frac{\text{dr}}{\text{dt}}=0.01?$

Variable $x$ and $y$ are related by equation $x =$ $\int\limits_0^y {\frac{{dt}}{{\sqrt {1 + {t^2}} }}} $ . The value of $\frac{{{d^2}y}}{{d{x^2}}}$ is equal to

If $x = {t^2}$, $y = {t^3}$, then ${{{d^2}y} \over {d{x^2}}} =$