Question

The positive value of the determinant of the matrix A, whose $\text{Adj}(\text{Adj}(A))=\left(\begin{array}{ccc}14& 28& -14\\ -14& 14& 28\\ 28& -14& 14\end{array}\right)$, is_____.

Answer 1

Answer: 14

$\text{Adj}(\text{Adj}A)=\left[\begin{array}{ccc}14& 18& -14\\ -14& 14& 28\\ 28& -14& 14\end{array}\right]$
$|\text{Adj}(\text{Adj}A)|=\left[\begin{array}{ccc}14& 28& -14\\ -14& 14& 28\\ 28& -14& 14\end{array}\right]=14\times 14\times 14\left|\begin{array}{ccc}1& 2& -1\\ -1& 1& 2\\ 2& -1& 1\end{array}\right|$
$=(14{)}^3[3-2(-5)-1(-1)]=(14{)}^3[14]=(14{)}^4$
$|A{|}^4=(14{)}^4\Rightarrow |A|=14$