Applications of determinant

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Question

$Let P, Q and R be invertible matrices of order three such$ $that A = P Q^{- 1}, B = Q R^{- 1} and$
$C = R P^{- 1}$ $Then the value of$ $\det (A B C + B C A + C A B) is$

Moderate

Solution

$We have A = P Q^{- 1}, B = Q R^{- 1}, C = R P^{- 1}$
$\begin{array}{l} ∴ A B C = I, B C A = I, C A B = I \\ ∴ \det . (A B C + B C A + C A B) = \det . (3 I) \\ = 3^{3} \times \det . (I) = 27 \end{array}$

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