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Questions
Evaluate the following determinants:
`|(x - 1, x, x - 2),(0, x - 2, x - 3),(0, 0, x - 3)| = 0`
Solve the following equation:
`|(x -1, x, x - 2),(0, x - 2, x - 3),(0, 0, x - 3)|` = 0
Solution 1
`|(x - 1, x, x - 2),(0, x - 2, x - 3),(0, 0, x - 3)| = 0`
∴ `(x - 1) |(x - 2, x - 3),(0, x - 3)| - x|(0, x - 3),(0, x - 3)| + (x - 2)|(0, x - 2),(0, 0)|` = 0
∴ (x - 1)[(x - 2)(x - 3) - 0] - x(0 - 0) + (x - 2)(0 - 0) = 0
∴ (x - 1)(x - 2)(x - 3) = 0
∴ x - 1 = 0 or x - 2 = 0 or x - 3 = 0
∴ x = 1 or x = 2 or x = 3
Solution 2
`|(x -1, x, x - 2),(0, x - 2, x - 3),(0, 0, x - 3)|` = 0
Applying R2 → R2 – R3, we get
`|(x -1, x, x - 2),(0, x - 2, 0),(0, 0, x - 3)|` = 0
∴ (x – 1)[(x – 2)(x – 3) – 0] – x(0 – 0) + (x – 2)(0 – 0) = 0
∴ (x – 1)(x – 2)(x – 3) = 0
∴ x – 1 = 0 or x – 2 = 0 or x – 3 = 0
∴ x = 1 or x = 2 or x = 3
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