Without expanding, show that the value of each of the following determinants is zero : |(0,x,y)(-x,0,z)(-y,-z,0)| ​

Question

Without expanding, show that the value of each of the following determinants is zero :

\(\begin{vmatrix} 0& x & y \\[0.3em] -x & 0 & z \\[0.3em] -y & -z &0 \end{vmatrix}\)

Answer 1

Let Δ =\(\begin{vmatrix} 0& x & y \\[0.3em] -x & 0 & z \\[0.3em] -y & -z &0 \end{vmatrix}\) 

Multiplying C₁, C₂ and C₃ with z, y and x respectively we get,

⇒ Δ = \((\frac{1}{xyz})\)\(\begin{vmatrix} 0& xy & yx \\[0.3em] -xz & 0 & zx \\[0.3em] -yz & -zy &0 \end{vmatrix}\) 

Now, 

Taking y, x and z common from R₁,R₂ and R₃ gives,

⇒ Δ = \((\frac{1}{xyz})\)\(\begin{vmatrix} 0& x & x \\[0.3em] -z & 0 & z \\[0.3em] -y & -y &0 \end{vmatrix}\) 

Applying C₂ → C₂ – C₃ gives,

⇒ Δ = \((\frac{1}{xyz})\)\(\begin{vmatrix} 0& x & x \\[0.3em] -z & -z & z \\[0.3em] -y & -y &0 \end{vmatrix}\) 

As, 

C₁ = C₂, 

Therefore determinant is zero.