Question

| x y z |
| x-y y-z z-x | =x³+y³+z³-3xyz
| y+z z+x x+y |
,prove it using theorems

Answer 1

Answer:

$\bf\left|\begin{array}{ccc}x&y&z\\x-y&y-z&z-x\\y+z&z+x&x+y\end{array}\right|=x^3+y^3+z^3-3xyz$

Step-by-step explanation:

$\left|\begin{array}{ccc}x&y&z\\x-y&y-z&z-x\\y+z&z+x&x+y\end{array}\right|$

$=\left|\begin{array}{ccc}x&y&z\\-y&-z&-x\\y+z&z+x&x+y\end{array}\right|$ $R_2\implies\:R_2-R_1$

$=\left|\begin{array}{ccc}x&y&z\\-y&-z&-x\\z&x&y\end{array}\right|$ $R_3\implies\:R_3+R_2$

Expanding along first row

$=x(-yz+x^2)-y(-y^2+xz)+z(-xy+z^2)$

$=-xyz+x^3+y^3-xyz-xyz+z^3$

$=x^3+y^3+z^3-3xyz$

$\implies\:\boxed{\bf\left|\begin{array}{ccc}x&y&z\\x-y&y-z&z-x\\y+z&z+x&x+y\end{array}\right|=x^3+y^3+z^3-3xyz}$