Question

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

Answer 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

Answer 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