<s>[INST] Solving a determinant using properties of a determinant 
$$\begin{vmatrix}
  y+z & x & x\\
  y & z+x & y\\
  z & z & x+y
 \end{vmatrix}=k(xyz)$$
   Find the value of $k$.

I solved this question by substituting $x=y=z=1$ and then expanding the determinant to get $k=4$ which is the correct answer.
Is there any method other than substitution and expanding the determinant right away?
 [/INST] Use the property that adding multiples of rows does not affect the determinant. Subtracting row 2 and 3 from row 1, we get
$$
\,\,\,\,\,\,\begin{vmatrix}
  y+z & x & x\\
  y & z+x & y\\
  z & z & x+y
 \end{vmatrix}\\=
\begin{vmatrix}
  0 & -2z & -2y\\
  y & z+x &y\\
  z & z & x+y
 \end{vmatrix}
$$
Add half row 1 to row 2 and 3,
$$
=\begin{vmatrix}
  0 & -2z & -2y\\
  y & x &0\\
  z & 0 & x
 \end{vmatrix}
$$
And now the rule of Sarrus will quickly get you $4xyz$.
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