Using Properties of determinants, prove that |(1, 1, 1+3x), (1+3y, 1, 1), (1, 1+3z, 1)|

Question

Using Properties of determinants, prove that

\(\begin{vmatrix} 1 & 1 & 1+3x\\ 1+3y & 1 & 1 \\ 1 & 1+3z & 1\end{vmatrix}\) = 9(3xyz + xy + yz + zx)

Answer 1

Let Δ = \(\begin{vmatrix} 1 & 1 & 1+3x\\ 1+3y & 1 & 1 \\ 1 & 1+3z & 1\end{vmatrix}\)

On applying R₂ → R₂ - R₃ and R₃ → R₃ - R₁

Δ = \(\begin{vmatrix} 1 & 1 & 1+3x\\ 3y & -3z & 0 \\ 1 & 3z & -3x\end{vmatrix}\)

On applying along R₁, we get

Δ = 9xy - 0] + (1 + 3x) (9yz - 0)

= 9zx + 9xy + 9yz + 27xyz = 9(3xyz + yz + zx + xy)

= 9(3xyz + yz + zx + xy)

= 9(3xyz + xy + yz + zx)

Hence proved