Question

If $A=\left[\begin{array}{ccc}-1& -2& -2\\ 2& 1& -2\\ 2& -2& 1\end{array}\right]$, Then $adj\left(A\right)$ equals

• A. $A$
• B. $A^T$
• C. $3A$
• D. $3A^T$

Answer 1

Answer: (D)

$3A^T$

Given $A=\left[\begin{array}{ccc}-1& -2& -2\\ 2& 1& -2\\ 2& -2& 1\end{array}\right]$
$A^T=\left[\begin{array}{ccc}-1& 2& 2\\ -2& 1& -2\\ -2& -2& 1\end{array}\right]$
Now, $adjA=C^T={\left[\begin{array}{ccc}-3& -6& -6\\ 6& 3& -6\\ 6& -6& 3\end{array}\right]}^T$
$\Rightarrow adjA=\left[\begin{array}{ccc}-3& 6& 6\\ -6& 3& -6\\ -6& -6& 3\end{array}\right]$
$\Rightarrow adjA=3\left[\begin{array}{ccc}-1& 2& 2\\ -2& 1& -2\\ -2& -2& 1\end{array}\right]$
$\Rightarrow adjA=3A^T$