If ax + cy + bz = X, cx + by + az = Y, bx + ay + cz = Z, show that :
$(a^{2} + b^{2} + c^{2} - b c - c a - a b) (x^{2} + y^{2} + z^{2}$ $- y z - z x - x y)$
$= (X^{2} + Y^{2} + Z^{2} - Y Z - Z X - X Y)$

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Solution

$(a^{2} + b^{2} + c^{2} - b c - c a - a b) (x^{2} + y^{2} + z^{2} - y z - z x - x y)$
$= (a + b ω + c ω^{2}) (a + b ω^{2} + c ω) (x + y ω + z ω^{2}) (x + y ω^{2} + z ω)$
$= [(a + b ω + c ω^{2}) (x + y ω + z ω^{2})] [(a + b ω^{2} + c ω) (x + y ω^{2} + z ω)]$
$= (a x + c y ω^{3} + b z ω^{3} + c x ω^{2} + b y ω^{2} + a z ω^{2} + b x ω + a y ω + c z ω^{4}) \times (a x + c y ω^{3} + b z ω^{3} + c x ω + b y ω^{4} + a z ω + b x ω^{2} + a y ω^{2} + c z ω^{2})$
$[(a x + c y + b z) (c x + b y + a z) ω^{2} + (b x + a y + c z) ω] \times [(a x + c y + b z) (c x + b y + a z) ω + (b x + a y + c z) ω^{2}]$
$(X + Y ω^{2} + Z ω) (X + Y ω + Z ω^{2})$
$(X^{2} + Y^{2} + Z^{2} - Y Z - Z X - X Y)$

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