Question

If x + y + z = 0, prove that |(xa yb zc), (yc za xb), (zb xc ya)| = xyz|(a b c), (c a b), (b c a)|

Answer 1

Answer:

The proof is explained below.

Step-by-step explanation:

Given x + y + z = 0, we have to prove that

$\begin{vmatrix}xa &yb&zc\\yc &za&xb\\zb &xc&ya \end{vmatrix}=xyz\begin{vmatrix}a&b &c\\c&a&b\\b&c&a \end{vmatrix}$

As we know, x+y+z=0 ⇒ $x^3+y^3+z^3=3xyz$

Now, taking L.H.S

$\begin{vmatrix}xa &yb&zc\\yc &za&xb\\zb &xc&ya \end{vmatrix}$

$=xa(a^2yz-x^2bc)-yb(y^2ac-b^2xz)+zc(c^2xy-z^ab)\\\\=xyza^3-x^3abc-y^3abc+b^3xyz+c^3xyz-z^3abc\\\\=xyz(a^3+b^3+c^3)-abc(x^3+b^3+c^3)\\\\=xyz(a^3+b^3+c^3)-abc(3xyz)\\\\=xyz(a^3+b^3+c^3-3abc)\\\\=xyz\begin{vmatrix}a&b &c\\c&a&b\\b&c&a \end{vmatrix}$

hence Proved.