Given that A is a square matrix of order 3 and \(|adj \cdot A| = 8\).

We know that \(adj(A) = |A|A^{-1}\), so \(|adj(A)| = |A|^{n-1}\) where n is the order of the matrix.

Also, \(|adj \cdot A| = |adj(A)| \cdot |A|\)

Since A is of order 3, \(|adj(A)| = |A|^{3-1} = |A|^2\)

Therefore, \(|adj \cdot A| = |A|^2 \cdot |A| = |A|^3\)

Given \(|adj \cdot A| = 8\), so \(|A|^3 = 8\)

Taking the cube root of both sides, we get \(|A| = \sqrt[3]{8} = 2\)

We need to find the value of \(|A^T|\). We know that the determinant of a transpose of a matrix is equal to the determinant of the original matrix, i.e., \(|A^T| = |A|\)

Therefore, \(|A^T| = 2\)