Forming basis in Linear Algebra Looking for some help with the following question. For which value or values of $k$ do the vectors below form a basis of $ \Bbb R^4$. 
$\begin{bmatrix}
1&0&0&2  
\\0&1&0&3   
\\0&0&1&4
\\2&3&4&k
\end{bmatrix}$
My thinking is that the columns of this matrix are linearly independent if and only if the matrix is invertible If and only if the determinant of the matrix $\ne0$
So looking for some help with this question.  Would I have to calculate the determinant of this matrix to see what values of $k$ would form a basis of $ \Bbb R^4$. And if so wouldn't there be more than one possible value of $k$ that would make this true? Looking for some help.

Hint:
$$ \begin{vmatrix}
1&0&0&2  
\\0&1&0&3   
\\0&0&1&4
\\2&3&4&k
\end{vmatrix} = \begin{vmatrix}
1&0&0&2  
\\0&1&0&3   
\\0&0&1&4
\\0&3&4&k-2(2)
\end{vmatrix} = \begin{vmatrix}
1&0&0&2  
\\0&1&0&3   
\\0&0&1&4
\\0&0&4&k-2(2)-3(3)
\end{vmatrix}$$
Are you able to finish the working?
