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Home  >>  JEEMAIN and AIPMT  >>  Mathematics  >>  Class12  >>  Determinants
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If $A$ is square matrix of order $n\times n$ then adj(adj(A)) is equal to

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

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Note: : The matrix formed by taking the transpose of the cofactor matrix of a given original matrix is called the adjoint of matrix A and is written as adj (A).
Since A(adj A)=|A|In|A|I_n
Replacing A by adj A,we get
(adj A)[adj(adj A)]=|adj A|$I_n$
$\Rightarrow A(adj A)[adj(adj A)]=A|A|^{n-1}I_n$
$|adj A|=|A|^{n-1}$
$\Rightarrow |A|I_n[adj(adj A)]=A|A|^{n-1}I_n$
$\Rightarrow adj(adj A)=|A|^{n-2}A$
Hence (C) is the correct answer.
answered Apr 25, 2014 by sreemathi.v
edited Jul 11, 2014 by balaji.thirumalai

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