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Question Number 183863 by Michaelfaraday last updated on 31/Dec/22

	Answered by qaz last updated on 31/Dec/22
	$(d/dθ){ determinant (( ((A,B),(C,D) ), ((E,F),(G,H) )))} =(d/dθ){ determinant ((A,B),(C,D))∙ determinant ((E,F),(G,H))} = determinant ((A,B),(C,D))^′ ∙ determinant ((E,F),(G,H))+ determinant ((A,B),(C,D))∙ determinant ((E,F),(G,H))^′ =( determinant (((A′),(B′)),(C,D))+ determinant ((A,B),((C′),(D′))))∙ determinant ((E,F),(G,H))+ determinant ((A,B),(C,D))∙( determinant (((E′),(F′)),(G,H))+ determinant ((E,F),((G′),(H′))))$
	$\frac{d}{d θ} {\| \begin{matrix} (\begin{matrix} A & B \\ C & D \end{matrix}) & (\begin{matrix} E & F \\ G & H \end{matrix}) \end{matrix} \|}$ $= \frac{d}{d θ} {\| \begin{matrix} A & B \\ C & D \end{matrix} \| \cdot \| \begin{matrix} E & F \\ G & H \end{matrix} \|}$ $= {\| \begin{matrix} A & B \\ C & D \end{matrix} \|}^{^{'}} \cdot \| \begin{matrix} E & F \\ G & H \end{matrix} \| + \| \begin{matrix} A & B \\ C & D \end{matrix} \| \cdot {\| \begin{matrix} E & F \\ G & H \end{matrix} \|}^{^{'}}$ $= (\| \begin{matrix} A^{'} & B^{'} \\ C & D \end{matrix} \| + \| \begin{matrix} A & B \\ C^{'} & D^{'} \end{matrix} \|) \cdot \| \begin{matrix} E & F \\ G & H \end{matrix} \| + \| \begin{matrix} A & B \\ C & D \end{matrix} \| \cdot (\| \begin{matrix} E^{'} & F^{'} \\ G & H \end{matrix} \| + \| \begin{matrix} E & F \\ G^{'} & H^{'} \end{matrix} \|)$
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