Problem 1604

1. Identify the structure of the matrix:
The given matrix is a $4 \times 4$ skew-symmetric matrix, where the elements satisfy the property $A_{ij} = -A_{ji}$ and $A_{ii} = 0$ for all $i$.

$$A = \begin{pmatrix} 0 & a & b & c \\ -a & 0 & d & e \\ -b & -d & 0 & f \\ -c & -e & -f & 0 \end{pmatrix}$$

2. Properties of skew-symmetric matrices:
For any skew-symmetric matrix $A$ of odd order, the determinant is zero. For even order, the determinant is the square of the Pfaffian of the matrix. The Pfaffian of a $4 \times 4$ skew-symmetric matrix can be computed as follows:

$$\text{Pf}(A) = af - be + cd$$

3. Compute the determinant using the Pfaffian:
The determinant of a $4 \times 4$ skew-symmetric matrix $A$ is given by:

$$\det(A) = (\text{Pf}(A))^2$$

Substituting the Pfaffian:

$$\det(A) = (af - be + cd)^2$$

4. Non-negativity of the determinant:
Since the square of any real number is non-negative, we have:

$$(af - be + cd)^2 \geq 0$$

Therefore, the determinant of the given matrix is non-negative.

$$\boxed{(af - be + cd)^2}$$