1. Tardigrade
  2. Question
  3. Mathematics
  4. Let a, b, c be any real numbers. Suppose that there are real numbers x, y, z not all zero such that x=c y+b z, y=a z+c x and z=b x+a y then a2+b2+c2+2 a b c is equal to

Question Error Report

Thank you for reporting, we will resolve it shortly

Back to Question

Q. Let $a, b, c$ be any real numbers. Suppose that there are real numbers $x, y, z$ not all zero such that $x=c y+b z, y=a z+c x$ and $z=b x+a y$ then $a^2+b^2+c^2+2 a b c$ is equal to

Determinants

Solution:

The equation system $x - cy - bz = 0$
$cx - y + az = 0$
$bx + ay - z = 0$
have non trivial solution if $\begin{vmatrix}1 & - c & - b \\ c & -1 & a \\ b & a & -1\end{vmatrix}=0$
$\text { Expand by } R_1, 1\left(1-a^2\right)+c(-c-a b)-b(a c+b)=0 $
$1-a^2-c^2-a b c-a b c-b^2=0 $
$a^2+b^2+c^2+2 a b c=1$