Question

Without actual expansion show that the value of the determinant `|(5,5^2,5^3),(5^2,5^3,5^4),(5^4,5^5,5^6)|` is zero.

Answer 1

Taking 5, 5² and 5⁴ common from R¹, R² and R³ respectively we get

`|(5,5^2,5^3),(5^2,5^3,5^4),(5^4,5^5,5^6)| = 5 xx 5^2 xx 5^4 |(1,5,5^2),(1,5,5^2),(1,5,5^2)|`

= 5 × 5^2 × 5^4 × 0 = 0   ....[R₁ ≡ R₂ ≡ R₃]