Question

If the matrix $A=\left|\begin{array}{ccc}1& 3& 1\\ -1& 2& -3\\ 0& 1& 2\end{array}\right|$ then $adj\left(adjA\right)$ is equal to

• A. $\left|\begin{array}{ccc}12& 36& 12\\ -12& 24& -36\\ 0& 12& 24\end{array}\right|$
• B. $\left|\begin{array}{ccc}12& 26& 12\\ 24& 36& -36\\ 0& 12& -24\end{array}\right|$
• C. $\left|\begin{array}{ccc}12& -12& 36\\ 24& -24& -36\\ 0& 12& 24\end{array}\right|$
• D. None of these

Answer 1

The correct option is A $\left|\begin{array}{ccc}12& 36& 12\\ -12& 24& -36\\ 0& 12& 24\end{array}\right|$

$A=\left|\begin{array}{ccc}1& 3& 1\\ -1& 2& -3\\ 0& 1& 2\end{array}\right|$

$⟹|A|=1\times (4+3)-3\times (-2+0)+1\times (-1-0)$
$=7+6-1=12$
So, $adj\left(adjA\right)=|A{|}^{n-2}A$
$=(12{)}^{3-2}A=12A$
$=12\left|\begin{array}{ccc}1& 3& 1\\ -1& 2& -3\\ 0& 1& 2\end{array}\right|$
$=\left|\begin{array}{ccc}12& 36& 12\\ -12& 24& -36\\ 0& 12& 24\end{array}\right|$

Hence, option A is correct.