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## Homework Statement

show that [itex]\det(\underline{\bf{A}})\det(\underline{\bf{B}}) = \det(\underline{\bf{AB}})[/itex]

## Homework Equations

\begin{align*}

&\underline{\bf{A}} = A_{ij} \underline{e}_i \otimes \underline{e}_j \\

&\underline{\bf{B}} = B_{mn} \underline{e}_m \otimes \underline{e}_n \\

&\underline{\bf{A}}\underline{\bf{B}} = A_{ij}B_{mn} \underline{e}_i \otimes \underline{e}_n \\

& \det({\underline{\bf{A}}})=\epsilon_{pqr}A_{p1}A_{q2}A_{r3} \\

& \det({\underline{\bf{B}}})=\epsilon_{xyz}B_{x1}B_{y2}B_{z3} \\

& \det({\underline{\bf{A}}})(\det{\underline{\bf{B}}})=\epsilon_{pqr}A_{p1}A_{q2}A_{r3}\epsilon_{xyz}B_{x1}B_{y2}B_{z3}

\end{align*}

## The Attempt at a Solution

\begin{align*}

&\underline{\bf{C}}={\underline{\bf{A}}}{\underline{\bf{B}}} \\

& \underline{\bf{C}} = A_{im}B_{mj} \underline{e}_i \otimes \underline{e}_j \\

& C_{ij} = A_{im}B_{mj} \\

& \det(\underline{\bf{AB}}) = \det(\underline{\bf{C}}) = \epsilon_{stu}C_{s1}C_{t2}C_{u3} \\

& C_{s1} = A_{sm}B_{m1} \\

& C_{t2} = A_{tn}B_{n2} \\

& C_{u3} = A_{uk}B_{k3} \\

& \det(\underline{\bf{AB}}) =\det(\underline{\bf{C}}) = \epsilon_{stu} A_{sm}A_{tn}A_{uk}B_{m1}B_{n2}B_{k3} \\

& \det(\underline{\bf{AB}}) =\det(\underline{\bf{C}}) = \epsilon_{pqr} A_{px}A_{qy}A_{rz}B_{x1}B_{y2}B_{z3}

\end{align*}

Which means I need to show that [itex]A_{px}A_{qy}A_{rz}=\epsilon_{xyz} A_{p1}A_{q2}A_{r3}[/itex]. That's where I'm stuck. Any hints?

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