I am just trying to work through this example, however I do not seem to get the same expression for the characteristic equation, no matter what I try. Hoping someone could help?
$det \begin{vmatrix} a_{12}-1-\lambda & a_{12}(a_{12}-1) \\ \rho a_{21}(a_{21}-1) & \rho(a_{21}-1)-\lambda \end{vmatrix} $
I should get
$ \lambda^{2} + ( 1-a_{12}+ \rho(1-a_{21}))\lambda + \rho(1-a_{21})(1-a_{12})(1-a_{12}a_{21}) $
However I only can get the first and second terms, not the constant term!
$ (a_{12}-1-\lambda)(\rho a_{21}-\rho-\lambda) - a_{12}(a_{12}-1)(\rho a_{21}(a_{21}-1) = 0 $
$ \rho a_{12}a_{21} - a_{12}\rho - a_{12}\lambda - \rho a_{21} + \rho + \lambda - \rho a_{21}\lambda + \rho\lambda + \lambda^{2}$
$ \lambda^{2} -a_{12} \lambda + \lambda -\rho a_{21} \lambda + \rho \lambda $
$\lambda^{2} + ( 1-a_{12} -\rho a_{21} + \rho) \lambda$
$ \lambda^{2} + ( 1-a12+ \rho(1-a21))\lambda $
However the constant terms I can't seem to do much with,
I am left with $ \rho a_{12}a_{21} - a_{12} \rho - \rho a_{21} + \rho - a_{12}(a_{12}-1) \rho a_{21}(a_{21}-1) $ Which i ccan seem to factor, caan someone please help me with this thank you..
The determinant is $$ ((a_{12}-1)-\lambda)(\rho(a_{21}-1)-\lambda)-\rho a_{21}(a_{21}-1)a_{12}(a_{12}-1) $$ which becomes $$ \rho(a_{12}-1)(a_{21}-1)-\rho a_{12}a_{21}(a_{12}-1)(a_{21}-1) -\lambda(a_{12}-1-\rho(a_{21}-1))+\lambda^2 $$ The constant term is $$ \rho(a_{12}-1)(a_{21}-1)(1-a_{12}a_{21}) $$