Compute the value of a 2×2 determinant with first row [3, 5] and second row [7, 2]; use the formula det = ad − bc and select the correct result.

Introduction / Context:Determinants are fundamental in linear algebra and circuit analysis (for example, in Cramer's rule to solve mesh or node equations). A 2×2 determinant is simple to evaluate and provides insight into invertibility and orientation of linear transformations.

Given Data / Assumptions:

• Matrix entries: first row (a, b) = (3, 5); second row (c, d) = (7, 2).
• All quantities are real numbers.
• Standard 2×2 determinant formula applies.

Concept / Approach:

For A = [[a, b], [c, d]], the determinant is computed as det(A) = ad − bc. The sign (positive or negative) indicates whether the associated linear mapping preserves or reverses orientation.

Step-by-Step Solution:

Verification / Alternative check:

Swap the two columns; the determinant would flip sign to +29. Since our column order is [3,5] and [7,2], the correct signed value is −29.

Why Other Options Are Wrong:

31 and 39 arise from adding products instead of subtracting. −31 results from an arithmetic slip (e.g., using 32 = 4 or 57 = 36 mistakenly).

Common Pitfalls:

Reversing the subtraction order, sign mistakes, or mis-multiplying 5*7. Always compute ad and bc separately and then subtract.

Final Answer:

–29