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Math Help - Exercise: An Application to Markov chains

  1. #1
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    Exercise: An Application to Markov chains

    If A 2x2, show that A^-1 = A^T if and only if :
    Exercise: An Application to Markov chains-untitled.png

    [Hint: If a^2+b^2=1, then a=cosθ, b= sinθ for some θ. Use cos(θ-)=cosθcosϕ+sinθsinϕ]
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  2. #2
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    Quote Originally Posted by sshh View Post
    If A 2x2, show that A^-1 = A^T if and only if :
    Click image for larger version. 

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    [Hint: If a^2+b^2=1, then a=cosθ, b= sinθ for some θ. Use cos(θ-)=cosθcosϕ+sinθsinϕ]

    If \displaystyle{A=\begin{pmatrix}a&b\\c&d\end{pmatri  x}\,,\,\,then\,\,A^t=\begin{pmatrix}a&c\\b&d\end{p  matrix} , so we get

    \displaystyle{A^{-1}=A^t\Longleftrightarrow I=AA^t=\begin{pmatrix}a^2+b^2&ac+bd\\ac+bd&c^2+d^2  \end{pmatrix}\Longrightarrow a^2+b^2=1=c^2+d^2\,,\,\,ac+bd=0 .

    Now use the huge hint you're given...

    Tonio
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  3. #3
    MHF Contributor Swlabr's Avatar
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    What have you done so far? Where are you stuck?

    Essentially, you solve the equation,

    \left( \begin{array}{ccc}<br />
a & b \\<br />
c & d \end{array} \right)<br />
\left( \begin{array}{ccc}<br />
a & c \\<br />
b & d \end{array} \right)=\left( \begin{array}{ccc}<br />
1 & 0 \\<br />
0 & 1 \end{array} \right)<br />
...
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