Using properties of determinants, prove that $$\left|\begin{array}{lll}x & x^{2} & yz \\ y & y^{2} & zx \\ z & z^{2} & xy\end{array}\right|=(x-y)(y-z)(z-x)(xy+yz+zx).$$

CBSE Class 12 · 2007 · DeterminantsLong Answer

Using properties of determinants, prove that

x y z x^{2} y^{2} z^{2} y z z x x y = (x - y) (y - z) (z - x) (x y + y z + z x) .

Answer: Proved

Multiply

R_{1}, R_{2}, R_{3}

x, y, z

respectively and divide outside by

x y z

Δ = \frac{1}{x y z} x^{2} y^{2} z^{2} x^{3} y^{3} z^{3} x y z x y z x y z = \frac{x y z}{x y z} x^{2} y^{2} z^{2} x^{3} y^{3} z^{3} 111 .

Rearranging columns and using the standard Vandermonde-type reduction with

R_{i} \to R_{i} - R_{j}

factors out

(x - y) (y - z) (z - x)

, leaving the symmetric factor

(x y + y z + z x)

Δ = (x - y) (y - z) (z - x) (x y + y z + z x) .

Proved.

CBSE Class 12 Board · 2007 · Subjective