Determinants

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The determinant |[1, 2, 3], [4, 5, 6], [7, 8, 9]| equals

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-1

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Subject: mathematics
Chapter: matrices and determinants
Topic: determinants
Difficulty: Medium
Year: 2025

Solution

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Observation: Notice that the differences between consecutive elements in each row are constant ( $2 - 1 = 1, 3 - 2 = 1$ ).
Row Operations: Apply $R_{2} \to R_{2} - R_{1}$ and $R_{3} \to R_{3} - R_{2}$ .
- New $R_{2} = [4 - 1, 5 - 2, 6 - 3] = [3, 3, 3]$
- New $R_{3} = [7 - 4, 8 - 5, 9 - 6] = [3, 3, 3]$
Identical Rows Property: Since two rows ( $R_{2}$ and $R_{3}$ ) are now identical, the value of the determinant is $0$ .
Alternative Expansion: $1 (45 - 48) - 2 (36 - 42) + 3 (32 - 35) = - 3 + 12 - 9 = 0$ .

This medium difficulty mathematics question is from the chapter matrices and determinants, covering the topic of determinants. It appeared in the 2025 exam.

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