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[tex]M = \left(\begin{array}{cc}4&-1 \\ 2&7 \end{array}\right)[/tex]

I need to show M^n as a formula of entries where n>0:

so say [tex]M = \left(\begin{array}{cc}4&-1 \\ 2&7 \end{array}\right)[/tex] is diagonalizable: [tex]M = SDS^{-1}[/tex]

then [tex]M^2=S^2 D^2 S^{-2} [/tex]

and... [tex]M^3=S^3 D^3 S^{-3} [/tex]

I can see from this that [tex]D^n = \left(\begin{array}{cc}\alpha^n&0 \\ 0&\beta^2 \end{array}\right)[/tex] assuming [tex]\alpha= \lambda_1[/tex] and that [tex]\beta= \lambda_2[/tex]

[tex]\alpha= 5[/tex] and [tex]\beta= 6[/tex]

[tex]D^n = \left(\begin{array}{cc}5^n&0 \\ 0&6^n \end{array}\right)[/tex]

so to find S...

[tex]\left(\begin{array}{cc}4-\alpha&-1 \\ 2&7-\beta \end{array}\right)[/tex]

[tex]\left(\begin{array}{cc}-1&-1 \\ 2&2 \end{array}\right)[/tex]

[tex]v_1= \left(\begin{array}{c}1\\ -1 \end{array}\right)[/tex]

[tex]\left(\begin{array}{cc}2&1 \\ 2&1 \end{array}\right)[/tex]

[tex]v_2= \left(\begin{array}{c}1\\ -2 \end{array}\right)[/tex]

[tex]S = \left(\begin{array}{cc}1&1 \\-1&-2 \end{array}\right)[/tex]

[tex]det(S)=-1[/tex]

[tex]S^{-1} = \left(\begin{array}{cc}-1&-1 \\1&2 \end{array}\right)[/tex]

this is where I am stuck, i dont know how to get S^n or S^-n

any ideas?

I need to show M^n as a formula of entries where n>0:

so say [tex]M = \left(\begin{array}{cc}4&-1 \\ 2&7 \end{array}\right)[/tex] is diagonalizable: [tex]M = SDS^{-1}[/tex]

then [tex]M^2=S^2 D^2 S^{-2} [/tex]

and... [tex]M^3=S^3 D^3 S^{-3} [/tex]

I can see from this that [tex]D^n = \left(\begin{array}{cc}\alpha^n&0 \\ 0&\beta^2 \end{array}\right)[/tex] assuming [tex]\alpha= \lambda_1[/tex] and that [tex]\beta= \lambda_2[/tex]

[tex]\alpha= 5[/tex] and [tex]\beta= 6[/tex]

[tex]D^n = \left(\begin{array}{cc}5^n&0 \\ 0&6^n \end{array}\right)[/tex]

so to find S...

[tex]\left(\begin{array}{cc}4-\alpha&-1 \\ 2&7-\beta \end{array}\right)[/tex]

[tex]\left(\begin{array}{cc}-1&-1 \\ 2&2 \end{array}\right)[/tex]

[tex]v_1= \left(\begin{array}{c}1\\ -1 \end{array}\right)[/tex]

[tex]\left(\begin{array}{cc}2&1 \\ 2&1 \end{array}\right)[/tex]

[tex]v_2= \left(\begin{array}{c}1\\ -2 \end{array}\right)[/tex]

[tex]S = \left(\begin{array}{cc}1&1 \\-1&-2 \end{array}\right)[/tex]

[tex]det(S)=-1[/tex]

[tex]S^{-1} = \left(\begin{array}{cc}-1&-1 \\1&2 \end{array}\right)[/tex]

this is where I am stuck, i dont know how to get S^n or S^-n

any ideas?

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