Find matrix determinant How do I reduce this matrix to row echelon form and hence find the determinant, or is there a way that I am unaware of that finds the determinant of this matrix without having to reduce it row echelon form given this is all I know and there exists no additional information.
$\left[
  \begin{array}{ccc}
   1+x  & 2 & 3 & 4 \\
    1 & 2+x & 3 & 4 \\
    1 & 2 & 3+x & 4 \\
    1 & 2 & 3 & 4+x \\
  \end{array}
\right]$

Assume $f(x)=\Delta$. It's a fourth degree polynomial.
$C_1\to C_1+C_2+C_3+C_4\implies x+10 $ is a factor.
$f(0)=0\implies 0$ is a root.// Repeated Rows
$f'(0)=0+0+0+0\implies 0$ is again repeated.// Repeated Rows
$f''(0)=0\implies 0$ is again repeated.// Repeated Rows for last time.
This all $\implies f(x)=x^3(x+10)$. Its obvious that coefficient of $x^4$ must be $1$.
For differentiation, differentiate one row and treat others as constant(like product rule). Add them all. It's pretty easy to see that they have same rows(without even writing them).
