2×2 determinant evaluation: The first row of a determinant is [10, 6] and the second row is [3, 5]. What is the value of this determinant (compute ad − bc)?

Introduction / Context:Determinants summarize key properties of matrices and are used in solving linear systems, computing inverses, and analyzing transformations. For 2×2 matrices, the determinant has a simple closed form: ad − bc. This question reinforces accurate arithmetic and sign handling for a basic case.

Given Data / Assumptions:

• Matrix rows: [10, 6] and [3, 5].
• Interpret as a 2×2 matrix [[10, 6], [3, 5]].
• Use the standard 2×2 determinant formula.

Concept / Approach:For a 2×2 matrix with entries a, b in the first row and c, d in the second row, the determinant value is ad − bc. Careful multiplication and subtraction yield the final scalar.

Step-by-Step Solution:

Verification / Alternative check:Cross-multiplication visualization: draw diagonals, product of the main diagonal minus product of the other diagonal. The arithmetic again gives 32, confirming the calculation.

Why Other Options Are Wrong:

• 18 or 50: These are intermediate products, not the required difference.
• −32: Incorrect sign; ad − bc is positive for these values.

Common Pitfalls:

• Reversing the subtraction order (bc − ad) and getting the wrong sign.
• Arithmetic slip in 6 * 3 or 10 * 5.

Final Answer:32