Determinant techniques: The 'expansion' method for evaluating determinants is best described as which of the following?

Introduction / Context:
Determinants can be computed with several techniques. For small matrices, manual expansion (such as Sarrus rule for 3×3, or cofactor expansion for 2×2 and 3×3) is straightforward and quick. For higher orders, systematic methods (row reduction, LU decomposition) are typically more efficient and less error-prone.

Given Data / Assumptions:

• We compare the 'expansion' approach (manual expansion along rows/columns or Sarrus for 3×3) to other methods.
• Goal: identify the range where expansion is practically suitable.

Concept / Approach:
Cofactor/expansion methods scale poorly as order grows because the number of terms increases rapidly. However, for second- and third-order determinants, expansion is concise and transparent, making it a preferred hand-calculation technique in courses and exams.

Step-by-Step Solution:

Verification / Alternative check:
Compare operation counts: cofactor expansion grows combinatorially, whereas elimination (row-reduction to triangular form) uses polynomial-time operations, explaining why expansion is recommended mainly for low orders.

Why Other Options Are Wrong:

• 'better than any other method': Not for high-order matrices.
• 'good for only one determinant': Nonsense; it applies to many but is manageable chiefly for small orders.
• 'more flexible than the cofactor method': Expansion is the cofactor method in essence for general determinants.

Common Pitfalls:

• Applying expansion to large matrices and making arithmetic errors due to the explosion of terms.

Final Answer:
good for second- and third-order determinants