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a)

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### Matrix $A$

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\$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 $\\mathrm{trace}(A) = \\var{trA},\\;\\;\\;\\mathrm{det}(A)=\\var{dA}$

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b)

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#### Finding the eigenvectors:

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1. $\\lambda=\\var{mnA}$

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We have the eigenspace is given by all $v=(x,y)$ such that $(\\simplify{A-{mnA}}I_2)v=(0,0)$ i.e.

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\$\\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}\$

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This gives the two equations:

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\$\\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).

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So putting $x=1$ in the first equation we get $y_1=\\var{-s*(a11-mnA)}$

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Hence the eigenvector we want is \$\\begin{pmatrix} 1 \\\\ \\var{-s*(a11-mnA)} \\end{pmatrix}\$

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2. $\\lambda=\\var{mxA}$

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In this case we have the equations:

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\$\\begin{eqnarray*} \\simplify[std]{{a11-mxA}x + {a12}y}&=&0\\\\ \\simplify[std]{{a21}x + {a22-mxA}y}&=&0 \\end{eqnarray*} \$

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Once again there is only one equation, so putting $x=1$ in the first equation we get $y_2=\\var{-s*(a11-mxA)}$

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Hence the eigenvector we want is \$\\begin{pmatrix} 1 \\\\ \\var{-s*(a11-mxA)} \\end{pmatrix}\$

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Find the eigenvalues and eigenvectors for the matrix $A$ where:
\$A=\\begin{pmatrix} \\var{a11}&\\var{a12}\\\\ \\var{a21}&\\var{a22} \\end{pmatrix}\\;\\;\\;\\;\\;\$

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Find the eigenvalues of $A$.

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Let $a_1$ be the least eigenvalue of $A,\\;\\;\\; a_1=\\;\\;$[[0]]

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Let $a_2$ be the greatest eigenvalue of $A,\\;\\; a_2=\\;\\;$[[1]]

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Find eigenvectors for $A$.

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Let $(1,y_1)$ be an eigenvector corresponding to $a_1,\\;\\;\\;\\;y_1=\\;\\;$[[0]]

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Let $(1,y_2)$ be an eigenvector corresponding to $a_2,\\;\\;\\;\\;y_2=\\;\\;$[[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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