Math Linear Algebra

Question

Find all values of k for which the vectors

[1, 1, 1, 1], [1, 2, 3, k], [1, 4, 9, k^2], [1, 8, 27, k^3]

are linearly dependent.

Answer 1

Vectors linearly dependent if det(A) = 0

WE expand this determinant by last column: -12 + 22k - 12k² + 2k³ = 0

k³ - 6k² + 11k - 6 = 0.

k³ - k² - 5k² + 5k + 6k - 6 = 0

(k - 1)(k² - 5k + 6) = 0; (k - 1)(k - 2)(k - 3) = 0

k = 1, k = 2, k = 3