LA2.1.8

Prove that the determinant of the matrix $\begin{bmatrix} y+z & z & y \\ z & z+x & x \\ y & x & x+y \end{bmatrix}=4xyz\,$

Let $A= \begin{vmatrix} y+z & z & y \\ z & z+x & x \\ y & x & x+y \end{vmatrix}\,$

Applying $C_1-(C_2+C_3)\,$

$A=\begin{vmatrix} (y+z)-(z+y) & z & y \\ z-(z+2x) & z+x & x \\ y-(2x+y) & x & x+y \end{vmatrix}\,$

$\begin{vmatrix} 0 & z & y \\ -2x & z+x & x \\ -2x & x & x+y \end{vmatrix}\,$

$-2x\begin{vmatrix} 0 & z & y \\ 1 & z+x & x \\ 1 & x & x+y \end{vmatrix}\,$

Applying $R_2\rightarrow (R_2-R_3)\,$ we get

$A=-2x\begin{vmatrix} 0 & z & y \\ 0 & z & -y \\ 1 & x & x+y \end{vmatrix}\,$

Expanding by the elements of C1, $A=-2x(-yz-yz)=4xyz\,$ =RHS