Determinant with unknown parameter. I'm given 4 vectors: $u_1, u_2, u_3$ and $u_4$. I'm going to type them in as points, because it will be easier to read, but think as them as column vectors. 
$$u_1 =( 5, λ, λ, λ), \hspace{10pt} u_2 =( λ, 5, λ, λ), \hspace{10pt} u_3 =( λ, λ, 5, λ), \hspace{10pt}u_4 =( λ, λ, λ, 5)$$
The task is to calculate the value of λ if the vectors where linearly dependent, as well as linearly independent. 
I managed to figure out that I could put them in a matrix, let's call it $A$, and set $det(A) = 0$ if the vectors should be linearly dependent, and $det(A) \neq 0$ if the vectors should be linearly independent. 
Some help to put me in the right direction would be great!

Your guess is correct. But what is the difficulty then? 
May be you are unable to get the determinant in a simpler way..?
$$\left|\begin{array}{cccc} 5&\lambda&\lambda&\lambda\\\lambda&5&\lambda&\lambda\\\lambda&\lambda&5&\lambda\\\lambda&\lambda&\lambda&5\end{array}\right|=(5-\lambda)^3\left|\begin{array}{rrrr} 1&0&0&\lambda\\-1&1&0&\lambda\\0&-1&1&\lambda\\0&0&-1&5\end{array}\right|=(5-\lambda)^3\left[\left|\begin{array}{rrr}1&0&\lambda\\-1&1&\lambda\\0&-1&5 \end{array}\right|+\left|\begin{array}{rrr}0&0&\lambda\\-1&1&\lambda\\0&-1&5 \end{array}\right|\right]$$
So, finally, the determinant is $$(5-\lambda)^3\left[(5+\lambda)+\lambda+\lambda\right]=(5-\lambda)^3(5+3\lambda)$$
