Question

If A is a square matrix of order n, then adj (adj A ) is equal to

• A. $|A{|}^{n-1}A$
• B. $|A{|}^{n}A$
• C. $|A{|}^{n-2}A$
• D. None of these

Answer 1

The correct option is C $|A{|}^{n-2}A$

For any square matrix X, we have X (adj X) = $|X|{]}_n$
Taking X = adj A, we get (adj A) [adj (adj A)] = $|adjA|I_n$
$\Rightarrow$ (adj A)[adj (adj A )] = $|adjA|I_n$
$\Rightarrow$ (adj A )[adj (adj A )] =$|A{|}^{n-1}{I}_n$
$[\because |adjA|=|A{|}^{n-1}]$
$Rightarrow$ (A adj A )[adj (adj A )] =$|A{|}^{n-1}A$
$[\because A{I}_n=A]$
$\left(|A|I_n\right)\left(adj\right(adjA)=|A{|}^{n-2}A$