Question

\( \left|\begin{array}{lll}x^{2} & x^{2}-(y-z)^{2} & y z \\ y^{2} & y^{2}-(z-x)^{2} & z x \\ z^{2} & z^{2}-(x-y)^{2} & x y\end{array}\right| \) \( =(x-y)(y-z)(z-x)(x+y+z)\left(x^{2}+y^{2}+z^{2}\right) \)

Answer 1

\(\begin{vmatrix}x^2&x^2-(y-z)^2&yz\\y^2&y^2-(z-x)^2&zx\\z^2&z^2-(x-y)^2&xy\end{vmatrix}\)

R₁→R₁ - R₂, R₂→R₂ - R₃

 = \(\begin{vmatrix}(x+y)(x-y)&(x+y)(x-y)-(x-y)(2z-x-y)&-z(x-y)\\(y+z)(y-z)&(y+z)(y-z)-(y-z)(2x-y-z)&-x(y-z)\\z^2&z^2-(x-y)^2& xy\end{vmatrix}\)

R₁→ R₁/(x - y),R₂→R₂/(y - z)

 = (x - y) (y - z)\(\begin{vmatrix}x+y&2(x + y + z)&-z\\y+z& 2(x +y+z)& -x\\z^2&z^2-(x-y)^2&xy\end{vmatrix}\) 

R₂→ R₂ - R₁

 = (x - y) (y - z) \(\begin{vmatrix}x+y&2(x+y+z)&-z\\z-x&0&z-x\\z^2&z^2-(x-y)^2& xy\end{vmatrix}\)

R₂ → \(\frac{R_2}{z-x}\)

 = (x - y) (y -z)(z - x)\(\begin{vmatrix}x+y&2(x+y+z)&-z\\1&0&1\\z^2&z^2-(x-y)^2&xy\end{vmatrix}\) 

C₃→ C₃ - C₁

= (x - y)(y - z) (z - x)\(\begin{vmatrix}x+y&2(x+y+z)&-(x+y+z)\\1&0&0\\z^2&z^2-(x-y)^2&xy-z^2\end{vmatrix}\) 

Expand determinant along C₁

= (x - y) (y - z) (z - x) (2(x + y + z) x (xy - z²) + (x + y + z)(z^2-(x - y)²))

 = -(x - y)(y - z) (z - x) (x + y + z)(2xy - 2z² + z² - x² - y² + 2xy)

 = -(x - y) (y - z) (z - x) (x + y + z)(-(x² + y² + z²) + 4xy)

 = (x - y) (y - z)(z - x) (x + y + z) (x² + y² + z² - 4xy)