Question

If A = $\left[\begin{array}{ccc}-1& -2& -2\\ 2& 1& -2\\ 2& -2& 1\end{array}\right]$ then Adj A =

• A. $A^T$
• B. $2A^T$
• C. $3A^T$
• D. $4A^T$

Answer 1

Answer: (C)

$3A^T$

$A=\left[\begin{array}{ccc}-1& -2& -2\\ 2& 1& -2\\ 2& -2& 1\end{array}\right]$

$A^T=\left[\begin{array}{ccc}-1& 2& 2\\ -2& 1& -2\\ -2& -2& 1\end{array}\right]$

Minor of $A=\left[\begin{array}{ccc}-3& +6& -6\\ -6& 3& 6\\ +6& 6& 3\end{array}\right]$

Co-factor$=\left[\begin{array}{ccc}-3& +6& -6\\ -6& 3& 6\\ +6& 6& 3\end{array}\right]\left[\begin{array}{ccc}+& -& +\\ -& +& -\\ +& -& +\end{array}\right]$

$=\left[\begin{array}{ccc}-3& -6& -6\\ +6& 3& -6\\ +6& -6& 3\end{array}\right]$

$=3\left[\begin{array}{ccc}-1& -2& -2\\ +2& 1& -2\\ +2& -2& 1\end{array}\right]$

Adj A$=$Transpose of co factor matrix$=3\left[\begin{array}{ccc}-1& +2& +2\\ -2& 1& -2\\ -2& -2& 1\end{array}\right]$

$=3A^T$

$\therefore AdjA=3A^T$

Option C is correct answer