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

To solve the given problem where the matrix Adj(Adj(A)) is defined as:

    Adj(Adj(A)) = 
    [
      [14, -14, 28],
      [28, 14, -14],
      [-14, 28, 14]
    ]

Step-by-Step Analysis

Understanding Adj(Adj(A))

The matrix Adj(Adj(A)), also known as the adjugate of the adjugate of A, is given. To find the determinant of A, we first calculate |Adj(Adj(A))|.

Calculating |Adj(Adj(A))|

The determinant of Adj(Adj(A)) can be written as:

    |Adj(Adj(A))| = |14 × 14 × 14|
                   × 
    [
      [1, -1, 2],
      [2, 1, -1],
      [-1, 2, 1]
    ]

Now, calculate the determinant of the smaller 3x3 matrix:

    |1 -1  2|
    |2  1 -1|
    |-1 2  1|

Expanding the Determinant

Using cofactor expansion:

    = 1 × (1 × 1 - (-1) × 2) - (-1) × (2 × 1 - (-1) × (-1)) + 2 × (2 × 2 - 1 × (-1))
    = 1 × (1 + 2) - (-1) × (2 - 1) + 2 × (4 + 1)
    = 1 × 3 + 1 × 1 + 2 × 5
    = 3 + 1 + 10
    = 14

Final Calculation for |A|

The determinant of Adj(Adj(A)) is given by:

    |Adj(Adj(A))| = (14)^3 × 14 = (14)^4

Since |Adj(Adj(A))| is related to |A| by the property of determinants:

    |Adj(Adj(A))| = |A|^4

Equating:

    (14)^4 = |A|^4

Taking the fourth root:

    |A| = 14