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 $64$

$A=\left[\begin{array}{ccc}1& 4& 9\\ 4& 9& 16\\ 9& 16& 25\end{array}\right]$
$A=\left|\begin{array}{ccc}1& 4& 9\\ 4& 9& 16\\ 9& 16& 25\end{array}\right|$
$C_3\to C_3-C_2,C_2\to C_2-C_1$
$|A|=\left|\begin{array}{ccc}1& 3& 5\\ 4& 5& 7\\ 9& 7& 9\end{array}\right|$
$C_3\to C_3-C_2,C_2\to C_2-C_1$
$|A|=\left|\begin{array}{ccc}1& 2& 2\\ 4& 1& 2\\ 9& -2& 2\end{array}\right|$
$\Rightarrow |A|=-8$
Now, $|adjA|=|A{|}^{n-1}$
$\Rightarrow |adjA|=(-8{)}^2=64$