Question

The positive value of the determinant of the matrix A, whose
$\text{Adj}(\text{Adj}(A)) = \begin{bmatrix} 14 & 28 & -14 \\\ -14 & 14 & 28 \\\ 28 & -14 & 14 \end{bmatrix}$
is ___.

Answer 1

The correct answer is 14
$|adj(adj(A))| = |A|^{2^2} = |A|^4$
$∴$ $|A|^4 = \left| \begin{matrix} 14 & 28 & -14 \\\ -14 & 14 & 28 \\\ 28 & -14 & 14 \end{matrix} \right|$
$= 14^3 \left| \begin{matrix} 1 & 2 & -1 \\\ -1 & 1 & 2 \\\ 2 & -1 & 1 \end{matrix} \right|$
$= (14)^3(3-2(-5)-1(-1))$
$= |A|^4 = (14)^4$
$⇒ |A| = 14$