The positive value of the determinant of the matrix A, whose Adj(Adj(A))=pmatrix14&28&-14-…

Question

The positive value of the determinant of the matrix $A$, whose $\operatorname{Adj}(\operatorname{Adj}(A))=\begin{pmatrix}14&28&-14\-14&14&28\28&-14&14\end{pmatrix}$, is\_\_\_\_

Answer: 14

Accepted Answer

For a $3\times3$ matrix, $|\operatorname{adj}(\operatorname{adj}A)|=|A|^4$. Factoring $14$ from every entry:
$$\begin{vmatrix}14&28&-14\-14&14&28\28&-14&14\end{vmatrix}=14^3\begin{vmatrix}1&2&-1\-1&1&2\2&-1&1\end{vmatrix}=14^3(1(1+2)-2(-1-4)+(-1)(1-2))=14^3(3+10+1)=14^3\times14=14^4.$$

So $|A|^4=14^4\Rightarrow|A|=14$ (positive value).

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