<s>[INST] Evaluate the determinant $\begin{vmatrix} y+z&z&y\\z&z+x&x\\y&x&x+y\end {vmatrix}$ Performing the operation $R_1\rightarrow R_1-R_2-R_3$
$$\begin{vmatrix} 0&-2x&-2x \\ y&z+x&x \\ z & x&x+y \end{vmatrix}$$
Pulling $-2x$ out and performing $C_2\rightarrow C_2-C_3$
$$-2x\begin{vmatrix} 0&0&1\\ y&z&x \\ z&-y&x+y \end{vmatrix}$$
$$=-2x(-y^2-z^2)$$
However, this is not given in the options, which are 
$4xyz$, $xyz$, $xy$, $x^3+y^3$
What am I doing wrong?
 [/INST] There is an error in your solution. The third row is not the same in the first step. The first step should be
$$\begin{vmatrix} 0&-2x&-2x \\ z&z+x&x \\ y & x&x+y \end{vmatrix}$$
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