Find all values of a for which the system has trivial solutions $$\begin{pmatrix}1+a & 1 & 1 & 1\\\ 1 & 1-a & 1 & 1\\1 & 1 & 1 & a\\1 & 2 & 2 & 1\\\end{pmatrix}$$
As the title says, find all values of a for which the system has trivial solutions. 
I have tried row reduction but I haven't gotten anywhere with it. So what I have done is calculate the determinant, which is:
2 a^3 - a
From what I understand system has trivial solutions if and only if det(A) != 0. 
So the solution for this problem would be every ${\rm I\!R}$ number except where det(A) == 0 ? 

Your determinant is correct, so we get some values to check
$$\det A = a \left(2 a^2-1\right) \implies a = 0, \pm \dfrac{1}{\sqrt{2}}$$
If $a = 0$, the RREF is
$$\begin{bmatrix}
 1 & 0 & 0 & 0 \\
 0 & 1 & 1 & 0 \\
 0 & 0 & 0 & 1 \\
 0 & 0 & 0 & 0 \\
\end{bmatrix}$$
If $a = \dfrac{1}{\sqrt{2}}$, the RREF is
$$\begin{bmatrix}
 1 & 0 & 0 & \sqrt{2}-1 \\
 0 & 1 & 0 & 1-\sqrt{2} \\
 0 & 0 & 1 & \frac{1}{\sqrt{2}} \\
 0 & 0 & 0 & 0 \\
\end{bmatrix}$$
If $a = -\dfrac{1}{\sqrt{2}}$, the RREF is
$$\begin{bmatrix}
 1 & 0 & 0 & -\sqrt{2}-1 \\
 0 & 1 & 0 & \sqrt{2}+1 \\
 0 & 0 & 1 & -\frac{1}{\sqrt{2}} \\
 0 & 0 & 0 & 0 \\
\end{bmatrix}$$
If $a \ne 0, a \ne \pm \dfrac{1}{\sqrt{2}}$, the RREF is
$$\begin{bmatrix}
 1 & 0 & 0 & 0 \\
 0 & 1 & 0 & 0 \\
 0 & 0 & 1 & 0 \\
 0 & 0 & 0 & 1 \\
\end{bmatrix}$$
Hopefully you can complete the question now.
