1. Theoretical Property: One of the key properties of determinants states that if any two rows (or columns) of a square matrix are identical, the determinant of that matrix is zero.
2. Proof by Row Operations: Consider a determinant $\Delta$ where Row 1 ($R_1$) and Row 2 ($R_2$) are identical. We can perform the row operation: $R_1 \to R_1 - R_2$.
Since $R_1$ and $R_2$ are identical, every element in the new $R_1$ will be:
$$\text{Element}_{R1} - \text{Element}_{R2} = 0$$
This results in a determinant where an entire row consists of zeros. According to another property of determinants, if all elements of a row or column are zero, the value of the determinant is
0.
3. Mathematical Illustration: Let $\Delta = \begin{vmatrix} a & b & c \\ a & b & c \\ d & e & f \end{vmatrix}$.
Expanding along the third row:
$$\Delta = d(bc - bc) - e(ac - ac) + f(ab - ab)$$
$$\Delta = d(0) - e(0) + f(0) = 0$$
Thus, the value is consistently zero regardless of the specific numbers involved.