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# If $A$ is a square matrix of order $n\times n$ then adj.(adj A) is equal to

$\begin{array}{1 1}(A)\;|A|^n&(B)\;|A|^{n-1}A\\(C)\;|A|^{n-2}A&(D)\;|A|^{n-3}A\end{array}$

Can you answer this question?

Since for a square matrix A,
$A(Adj.A)=|A|I_n$
Replacing A by adj A we get
$(adj A)(adj.(adj.A))=|adj.A|I_n$
$\Rightarrow |A|^{n-1}I_n$
$|adj.A|=|A|^{n-1}$
$\Rightarrow (Aadj.A)(adj.adj.A)=|A|^{n-1}A$
$\Rightarrow (|A|I_n)(adj.adj.A)=|A|^{n-1}A$
$\Rightarrow adj.(adj.A)=|A|^{n-2}A$
Hence (C) is the correct answer.
answered Apr 24, 2014