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  4. If A is a 3× 3 non-singular matrix, then det (A-1 adj A) is equal to

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Q. If A is a $ 3\times 3 $ non-singular matrix, then $ \det \,\,({{A}^{-1}}\,adj\,A) $ is equal to

J & K CETJ & K CET 2003

Solution:

det $ ({{A}^{-1}}\,adj\,\,A) $ =det $ ({{A}^{-1}})\,\det \,(adj\,A) $
$={{(\det \,A)}^{-1}}\,\,{{(\det \,\,A)}^{3-1}} $
$=\det \,A $