Linear algebra refresher – Ways to evaluate a 3×3 determinant Statement: “Third-order determinants are evaluated by the expansion method or by the cofactor method.”

Introduction / Context:In circuit theory, 3×3 determinants appear when solving three simultaneous equations (e.g., three-node nodal analysis with Cramer’s rule). Knowing standard evaluation methods speeds up hand calculations and checks.

Given Data / Assumptions:

• General 3×3 matrix with no special structure.
• Common textbook techniques include cofactor (Laplace) expansion and equivalent expansion by minors.
• Alternative mnemonics like Sarrus’ rule exist specifically for 3×3.

Concept / Approach:

“Expansion method” typically refers to expanding along a row or column using minors; the “cofactor method” is the same approach stated formally with signs (cofactors). Thus the statement is correct. One may also use Sarrus’ rule for 3×3, or convert to triangular form via elimination and multiply diagonal elements, but the assertion remains accurate as written.

Step-by-Step Solution:

Verification / Alternative check:

Cross-verify with Sarrus’ shortcut for 3×3 matrices to ensure numerical consistency in practice.

Why Other Options Are Wrong:

“False” contradicts standard linear algebra. Claiming “only Gaussian elimination is valid” ignores many correct analytical methods.

Common Pitfalls:

Forgetting the checkerboard sign pattern of cofactors; arithmetic slips when computing minors; misapplying Sarrus to non-3×3 cases.

Final Answer:

True.