Det(exp(A transpose)) is equal to the determinant of the matrix obtained by taking the transpose of the exponential of matrix A.

The Characteristics Equation of Matrix A

The characteristics equation of a matrix A is given by the equation:
p(λ) = |A - λI| = 0,
where p(λ) is the characteristic polynomial, A is the matrix, λ is the eigenvalue, and I is the identity matrix.

In this case, the given characteristics equation is:
A^3 - 3A^2 - 4A - 12I = 0.

Taking the transpose of both sides, we have:
(A^3 - 3A^2 - 4A - 12I)^T = 0.

The transpose of the identity matrix is itself, so we get:
(A^3)^T - (3A^2)^T - (4A)^T - (12I)^T = 0.

Since the transpose of a matrix product is the product of the transposes in reverse order, we have:
(A^T)^3 - (3A^T)^2 - (4A^T) - 12I = 0.

Simplifying, we get:
A^T^3 - 3A^T^2 - 4A^T - 12I = 0.

The Determinant of exp(A transpose)

Now, let's consider the determinant of exp(A transpose). We can write this as:
Det(exp(A transpose)) = |exp(A transpose)|.

Using the property that the determinant of a transpose is equal to the determinant of the original matrix, we have:
Det(exp(A transpose)) = |exp(A)|.

Since the determinant of a product is equal to the product of the determinants, we can further simplify:
Det(exp(A transpose)) = exp(Det(A)).

Therefore, the determinant of exp(A transpose) is equal to the exponential of the determinant of matrix A.

In conclusion, Det(exp(A transpose)) is equal to the exponential of the determinant of matrix A.