# Thread: Exercise: An Application to Markov chains

1. ## Exercise: An Application to Markov chains

If A 2x2, show that A^-1 = A^T if and only if :

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

2. Originally Posted by sshh
If A 2x2, show that A^-1 = A^T if and only if :

[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

3. What have you done so far? Where are you stuck?

Essentially, you solve the equation,

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