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જો $AA^T = I$ અને $C$ એ વિસંમિત શ્રેણિક છે તો $((A^T CA)^{50})^T$ મેળવો.
$A^{50}(C^T)^{50}(A^T)^{50}$
$A^TC^{50}A$
$-A^TC^{50}A$
$-AC^{50}A^T$
Solution
Given that $\mathrm{C}^{\mathrm{T}}=-\mathrm{C}$ and $\mathrm{AA}^{\mathrm{T}}=\mathrm{I}$
So
${\left( {{{\left( {{{\rm{A}}^{\rm{T}}}{\rm{CA}}} \right)}^{50}}} \right)^{\rm{T}}} = \mathop {{{\left[ {\left( {{{\rm{A}}^{\rm{T}}}{\rm{CA}}} \right)\left( {{{\rm{A}}^{\rm{T}}}{\rm{CA}}} \right)\left( {{{\rm{A}}^{\rm{T}}}{\rm{CA}}} \right) \ldots \left( {{{\rm{A}}^{\rm{T}}}{\rm{CA}}} \right)} \right]}^{\rm{T}}}}\limits_{50\,\,times} $
$=\left(\mathrm{A}^{\mathrm{T}} \mathrm{C}^{50} \mathrm{A}\right)^{\mathrm{T}}$
$\Rightarrow \mathrm{A}^{\mathrm{T}}\left(\mathrm{C}^{50}\right)^{\mathrm{T}} \mathrm{A}=\mathrm{A}^{\mathrm{T}}\left(\mathrm{C}^{\mathrm{T}}\right)^{50} \mathrm{A}=\mathrm{A}^{\mathrm{T}}(-\mathrm{C})^{50} \mathrm{A}=\mathrm{A}^{\mathrm{T}} \mathrm{C}^{50} \mathrm{A}$
