Linear algebra refresher – Evaluating a 2×2 determinant by cross-products Statement: “Second-order determinants are evaluated by subtracting the signed cross-products.”

Introduction / Context:Determinants are ubiquitous in circuit analysis (e.g., solving nodal equations via Cramer’s rule). Remembering how to compute a 2×2 determinant quickly is vital during timed exams and practical calculations.

Given Data / Assumptions:

• A general 2×2 determinant is |a b; c d|.
• We are using standard determinant definitions (no special numeric assumptions).
• “Signed cross-products” refers to the conventional ad − bc ordering with sign.

Concept / Approach:

The value of a 2×2 determinant is computed as the product of the main diagonal minus the product of the off-diagonal: det = ad − bc. This is precisely “subtracting the signed cross-products.” The sign matters; reversing terms would flip the determinant’s sign and change solutions when used in linear systems.

Step-by-Step Solution:

Verification / Alternative check:

Check with a simple numeric example, e.g., [[2, 3], [1, 4]] gives det = 24 − 31 = 8 − 3 = 5, matching the rule.

Why Other Options Are Wrong:

“False,” “True only if diagonal entries are positive,” and “True only for symmetric matrices” introduce conditions that are irrelevant; the formula holds universally for 2×2 matrices.

Common Pitfalls:

Accidental sign reversal (writing bc − ad), or mixing the rule with Sarrus’ method (which is for 3×3 determinants, not 2×2).

Final Answer:

True.