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)?
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A18
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B50
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C32
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D–32
Answer
Correct Answer: 32
Explanation
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:
Let a = 10, b = 6, c = 3, d = 5.Compute products: ad = 10 * 5 = 50; bc = 6 * 3 = 18.Evaluate determinant: ad − bc = 50 − 18 = 32.Thus, the determinant equals 32.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