a)
\n\\[A - \\lambda I_2 = \\begin{pmatrix} \\var{a11}-\\lambda & \\var{a12}\\\\ \\var{a21} & \\var{a22}-\\lambda \\end{pmatrix}\\]
Hence the characteristic polynomial $p(\\lambda)$ is: \\[\\begin{eqnarray*} \\mathrm{det}\\left(A-\\lambda I_2 \\right)&=&\\simplify[zeroTerm]{({a11}-lambda)({a22}-lambda)-{a12}*{a21}}\\\\ &=& \\simplify[std]{lambda^2-{trA}*lambda+{dA}}\\\\ &=&\\simplify[std]{(lambda-{a})(lambda-{b})} \\end{eqnarray*} \\]
We see that on solving $p(\\lambda)=0$ we get the eigenvalues:
\\[\\lambda_1=\\var{mnA},\\;\\;\\;\\lambda_2=\\var{mxA}\\]
Note: We could have found the characteristic polynomial by noting that for a 2 × 2 matrix $A$ then the characteristic polynomial is
\\[\\lambda^2-\\mathrm{trace}(A)+\\mathrm{det}(A)\\]
where $\\mathrm{trace}(A) = \\var{trA},\\;\\;\\;\\mathrm{det}(A)=\\var{dA}$
b)
\n1. $\\lambda=\\var{mnA}$
\nWe have the eigenspace is given by all $v=(x,y)$ such that $(\\simplify{A-{mnA}}I_2)v=(0,0)$ i.e.
\n\\[\\begin{pmatrix} \\var{a11-mnA}&\\var{a12}\\\\ \\var{a21}&\\var{a22-mnA} \\end{pmatrix}\\begin{pmatrix} x \\\\ y \\end{pmatrix} =\\begin{pmatrix} 0 \\\\ 0 \\end{pmatrix}\\]
\nThis gives the two equations:
\n\\[ \\begin{eqnarray*} \\simplify[std]{{a11-mnA}x + {a12}y}&=&0\\\\ \\simplify[std]{{a21}x + {a22-mnA}y}&=&0 \\end{eqnarray*} \\]
There is only one equation here as we see that the equations are the same (one is a multiple of the other).
So putting $x=1$ in the first equation we get $y_1=\\var{-s*(a11-mnA)}$
\nHence the eigenvector we want is \\[\\begin{pmatrix} 1 \\\\ \\var{-s*(a11-mnA)} \\end{pmatrix}\\]
\n2. $\\lambda=\\var{mxA}$
\nIn this case we have the equations:
\n\\[ \\begin{eqnarray*} \\simplify[std]{{a11-mxA}x + {a12}y}&=&0\\\\ \\simplify[std]{{a21}x + {a22-mxA}y}&=&0 \\end{eqnarray*} \\]
\nOnce again there is only one equation, so putting $x=1$ in the first equation we get $y_2=\\var{-s*(a11-mxA)}$
\nHence the eigenvector we want is \\[\\begin{pmatrix} 1 \\\\ \\var{-s*(a11-mxA)} \\end{pmatrix}\\]
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\\[ A=\\begin{pmatrix} \\var{a11}&\\var{a12}\\\\ \\var{a21}&\\var{a22} \\end{pmatrix}\\;\\;\\;\\;\\;\\]
Find the eigenvalues of $A$.
\nLet $a_1$ be the least eigenvalue of $A,\\;\\;\\; a_1=\\;\\;$[[0]]
\nLet $a_2$ be the greatest eigenvalue of $A,\\;\\; a_2=\\;\\;$[[1]]
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"$A,\\;B$ $2 \\times 2$ matrices. Find eigenvalues and eigenvectors of both. Hence or otherwise, find $B^n$ for largish $n$.
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