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  4. If adj B = A , |P| = |Q| = 1, then adj (Q-1BP-1) is

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Q. If $adj \, B = A , \left|P\right| = \left|Q\right| = 1$, then $adj \,(Q^{-1}BP^{-1})$ is

Determinants

Solution:

Given $adj \, B = A , \left|P\right| = \left|Q\right| = 1$
Consider , $adj \,\left(Q^{-1}\,BP^{-1}\right)$
$= \left(adj\, P^{-1}\right) \left(adj \,B\right) \left(adj \,Q^{-1}\right)$
$= \left(adj\,P\right)^{-1}\,A.\left(adj\,Q\right)^{-1}$
$= \left(P^{-1}\right)^{-1} A\left(Q^{-1}\right)^{-1}\quad\left(\because P^{-1} = \frac{1}{\left|P\right|} . adj\,P \right)$
$= PAQ .$