If A and B are non-singular matrices of same order, then which of the following is/are incorrect?

(a d j (A B))^{- 1} = (a d j B)^{- 1} (a d j A)^{- 1}

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(a d j A) (a d j A^{- 1}) = A^{2}

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(a d j A)^{- 1} = \frac{A}{| A |}

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(a d j A)^{- 1} A (a d j A) = A^{- 1}

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Solution

The correct option is D $(a d j A)^{- 1} A (a d j A) = A^{- 1}$
$A (a d j A) = (a d j A) A = | A | I$
 $\Rightarrow (a d j A)^{- 1} A (a d j A) = (a d j A)^{- 1} (a d j A) A$
$\Rightarrow (a d j A)^{- 1} A (a d j A) = A$
$A^{- 1} = \frac{adj A}{| A |}$
$a d j (A) = | A | A^{- 1}$
$(a d j A)^{- 1} = (| A | A^{- 1})^{- 1} = (| A |)^{- 1} (A^{- 1})^{- 1} = \frac{A}{| A |}$
$(a d j A) (a d j A^{- 1}) = (| A | A^{- 1}) (| A^{- 1} | (A^{- 1})^{- 1})$
$= | A | A^{- 1} | A |^{- 1} A = I$
or $(a d j A) (a d j A^{- 1}) = a d j (A^{- 1} A) = a d j I = I$
$(a d j A B)^{- 1} = (a d j B a d j A)^{- 1} = (a d j A)^{- 1} (a d j B)^{- 1}$

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