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.
$\left[
  \begin{array}{ccc}
   1+x  & 2 & 3 & 4 \\
    1 & 2+x & 3 & 4 \\
    1 & 2 & 3+x & 4 \\
    1 & 2 & 3 & 4+x \\
  \end{array}
\right]$

If you know how value of determinant is influenced by elementary row/column operations (see ProofWiki) then you could start by adding all other columns to the last one (which does not change the determinant) and the rest is relatively easy:
$$
\begin{vmatrix}
   1+x  & 2 & 3 & 4 \\
    1 & 2+x & 3 & 4 \\
    1 & 2 & 3+x & 4 \\
    1 & 2 & 3 & 4+x
\end{vmatrix}=
\begin{vmatrix}
   1+x  & 2 & 3 & 10+x \\
    1 & 2+x & 3 & 10+x \\
    1 & 2 & 3+x & 10+x \\
    1 & 2 & 3 & 10+x
\end{vmatrix}=
(x+10)
\begin{vmatrix}
   1+x  & 2 & 3 & 1 \\
    1 & 2+x & 3 & 1 \\
    1 & 2 & 3+x & 1 \\
    1 & 2 & 3 & 1
\end{vmatrix}=
(x+10)
\begin{vmatrix}
    x & 0 & 0 & 1 \\
    0 & x & 0 & 1 \\
    0 & 0 & x & 1 \\
    0 & 0 & 0 & 1
\end{vmatrix}=(x+10)x^3
$$
