Question

If $A=\left[\begin{array}{ccc}{1}^2& 2^2& 3^2\\ 2^2& 3^2& 4^2\\ 3^2& 4^2& 5^2\end{array}\right]$ then $|AdjA|=$

• A. $8$
• B. $16$
• C. $64$
• D. $128$

Answer 1

The correct option is C $64$

Consider the given matrix.

$A=\left[\begin{array}{ccc}{1}^2& 2^2& 3^2\\ 2^2& 3^2& 4^2\\ 3^2& 4^2& 5^2\end{array}\right]$

$A=\left[\begin{array}{ccc}1& 4& 9\\ 4& 9& 16\\ 9& 16& 25\end{array}\right]$

Therefore, the ad-joint of this matrix,

$adj\left(A\right)={\left[\begin{array}{ccc}\left|\begin{array}{cc}9& 16\\ 16& 25\end{array}\right|& -\left|\begin{array}{cc}4& 16\\ 9& 25\end{array}\right|& \left|\begin{array}{cc}4& 9\\ 9& 16\end{array}\right|\\ -\left|\begin{array}{cc}4& 9\\ 16& 25\end{array}\right|& \left|\begin{array}{cc}1& 9\\ 9& 25\end{array}\right|& -\left|\begin{array}{cc}1& 4\\ 9& 16\end{array}\right|\\ \left|\begin{array}{cc}4& 9\\ 9& 16\end{array}\right|& -\left|\begin{array}{cc}1& 9\\ 4& 16\end{array}\right|& \left|\begin{array}{cc}1& 4\\ 4& 9\end{array}\right|\end{array}\right]}^T$

$adj\left(A\right)={\left[\begin{array}{ccc}-31& 44& -17\\ 44& -56& 20\\ -17& 20& -7\end{array}\right]}^T$

Therefore,

$adj\left(A\right)=\left[\begin{array}{ccc}-31& 44& -17\\ 44& -56& 20\\ -17& 20& -7\end{array}\right]$

$\left|adj\left(A\right)\right|=-31(392-400)-44(-308+340)-17(880-952)$

$\left|adj\left(A\right)\right|=-31(-8)-44\left(32\right)-17(-72)$

$\left|adj\left(A\right)\right|=248-1408+1224$

$\left|adj\left(A\right)\right|=64$

Hence, the value of $\left|adj\left(A\right)\right|$ is $64$.