If \(ae+bg=cf+dh=p\), \(af+bh=ce+dg=q\), then what is \(\begin{vmatrix} a & b \\ c & d \end{vmatrix} \times \begin{vmatrix} e & f \\ g & h \end{vmatrix}\) equal to?
\(p^2-q^2\)
Let \(A=\begin{pmatrix} a & b \\ c & d \end{pmatrix}\) and \(B=\begin{pmatrix} e & f \\ g & h \end{pmatrix}\). Then \(AB=\begin{pmatrix} ae+bg & af+bh \\ ce+dg & cf+dh \end{pmatrix}=\begin{pmatrix} p & q \\ q & p \end{pmatrix}\) using the given conditions. Since \(\det(A)\cdot\det(B)=\det(AB)=p^2-q^2\), the required product equals \(p^2-q^2\).
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