× Close
Join the Ques10 Community
Ques10 is a community of thousands of students, teachers, and academic experts, just like you.
Join them; it only takes a minute
Sign up
Question: If a matrix A is given, show the following
0

$A= $ $ \left[ {\begin{array}{cc} 0 & -1 \\ 1 & 1 \\ \end{array} } \right] $

$e^{At}$=$ \left[ {\begin{array}{cc} cost & -sint \\ sint & cost \\ \end{array} } \right] $

Subject: Applied Mathematics 4

Topic: Matrices

Difficulty: Medium

m4m(64) • 80 views
ADD COMMENTlink
modified 11 weeks ago  • written 11 weeks ago by gravatar for manasahegde234 manasahegde23420
0

Characteristic equation of $\mid A-\lambda I \mid=0$

Therefore, $\lambda^2-(0)\lambda+\mid A \mid=0$

$\lambda^2+(0-(-1))=0$

$\lambda^2+1=0$

$\lambda=\pm i$

Let, $\phi(A)=e^At$

Consider, $\phi(A)=\alpha_1 A+\alpha_0 I$

$e^{At}=\alpha_1 A+\alpha_0 I $ (1)

$e^{\lambda t}=\alpha_1 \lambda+\alpha_0 $ (2)

Put $\lambda=i_1-1$ in (2), we get

$e^{it}=\alpha_1 i+\alpha_0$ (3)

$e^{-it}=-\alpha_1 i+\alpha_0$ (4)

adding (3) and (4) we get,

$e^{it}+e^{-it}=2\alpha_0$

$\alpha_0=\frac{e^{it}-e^{-it}}{2i}=sint$

Therefore, from (1) we get,

$e^{AT}=sintA+costI$

$e^{At}=sint\left[ {\begin{array}{cc} 0 & -1\\ 1 & 0 \end{array} } \right] $+$cost\left[ {\begin{array}{cc} 1 & 0\\ 0 & 1 \end{array} } \right] $=$\left[ {\begin{array}{cc} 0 & -sint\\ sint & 0 \end{array} } \right] $+$\left[ {\begin{array}{cc} cost & 0\\ 0 & cost \end{array} } \right] $

=$\left[ {\begin{array}{cc} cost & -sint\\ sint & cost \end{array} } \right] $

ADD COMMENTlink
written 11 weeks ago by gravatar for manasahegde234 manasahegde23420
Please log in to add an answer.


Use of this site constitutes acceptance of our User Agreement and Privacy Policy.