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Question Number 77565 by TawaTawa last updated on 07/Jan/20

Commented by MJS last updated on 07/Jan/20

$$\mathrm{I}{\mathrm{don}}^{\prime }\mathrm{t}\mathrm{think}\mathrm{that}{A}^B\mathrm{for}\mathrm{matrices}A,B\mathrm{is}$$$$\mathrm{defined}.\mathrm{I}\mathrm{only}\mathrm{know}{A}^n\mathrm{with}n\in \mathbb{N};A^{-n}\mathrm{is}$$$$\mathrm{only}\mathrm{possible}\mathrm{if}\mathrm{the}\mathrm{inverse}\mathrm{matrix}{A}^{-1}\mathrm{exists}$$$$\mathrm{and}\mathrm{is}\mathrm{then}\mathrm{defined}\mathrm{as}{\left(A^{-1}\right)}^n.A^{\frac{p}{q}}\mathrm{is}\mathrm{only}$$$$\mathrm{possible}\mathrm{if}A\mathrm{can}\mathrm{be}\mathrm{diagonalized}\mathrm{and}\mathrm{the}$$$$\mathrm{elements}\mathrm{of}\mathrm{the}\mathrm{diagonal}\mathrm{all}\mathrm{are}⩾0.\mathrm{we}\mathrm{find}$$$$\mathrm{the}\mathrm{diagonal}\mathrm{matrix}\mathrm{if}\mathrm{a}\mathrm{matrix}T\mathrm{exists}\mathrm{with}$$$$A=T\times D\times T^{-1}\mathrm{in}\mathrm{this}\mathrm{case}{A}^n=T\times D^n\times T^{-1}$$$$\mathrm{and}\mathrm{in}\mathrm{some}\mathrm{cases}{A}^{\frac{p}{q}}=T\times D^{\frac{p}{q}}\times T^{-1}$$

Commented by TawaTawa last updated on 07/Jan/20

$$\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}.\mathrm{I}\mathrm{appreciate}$$

Commented by TawaTawa last updated on 08/Jan/20

$$\mathrm{Sir}\mathrm{what}\mathrm{if}\mathrm{the}\mathrm{power}\mathrm{is}\mathrm{a}\mathrm{determinant}.$$

Commented by TawaTawa last updated on 08/Jan/20

$${\left[\begin{array}{c}7-3\\ 10-4\end{array}\right]}^{\left|\begin{array}{c}01\\ 3-2\end{array}\right|}$$

Commented by MJS last updated on 08/Jan/20

$$\det \left|\begin{array}{cc}0& 1\\ 3& -2\end{array}\right|=-3$$$${\left[\begin{array}{cc}7& -3\\ 10& -4\end{array}\right]}^{-1}=\left[\begin{array}{cc}-2& 3/2\\ -5& 7/2\end{array}\right]$$$${\left[\begin{array}{cc}-2& 3/2\\ -5& 7/2\end{array}\right]}^3=\left[\begin{array}{cc}-17/4& 21/8\\ -35/4& 43/8\end{array}\right]$$

Commented by TawaTawa last updated on 08/Jan/20

$$\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}$$

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