determinants

Question

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

RankGuru · Accepted Answer

1. **Observation**: Notice that the differences between consecutive elements in each row are constant ($2-1=1, 3-2=1$).
2. **Row Operations**: Apply $R_2 	o R_2 - R_1$ and $R_3 	o R_3 - R_2$.
   - New $R_2 = [4-1, 5-2, 6-3] = [3, 3, 3]$
   - New $R_3 = [7-4, 8-5, 9-6] = [3, 3, 3]$
3. **Identical Rows Property**: Since two rows ($R_2$ and $R_3$) are now identical, the value of the determinant is **$0$**.
4. **Alternative Expansion**:
   $1(45-48) - 2(36-42) + 3(32-35) = -3 + 12 - 9 = 0$.

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