If $a, b, c, \lambda \in \mathbb{N}$, then the least possible value of $\begin{vmatrix} a^2 + \lambda & ab & ac \\ ba & b^2 + \lambda & bc \\ ca & cb & c^2 + \lambda \end{vmatrix}$ is
Step-by-Step Solution
Key Concept: Row and column operations combined with factoring constants from rows/columns simplify determinant calculations.
Multiplying rows 1, 2, 3 by $a, b, c$ respectively and factoring $a, b, c$ from columns 1, 2, 3, the determinant becomes $(a^2 + b^2 + c^2 + \lambda)$ times a remaining determinant. Operating $R_1 \to R_1 + R_2 + R_3$ and factoring common terms, we obtain the final determinant value. Through row and column operations simplifying the structure, the determinant evaluates to $4$.
Correct Answer: 4