A matrix is given in echelon form This is my first time posting a question and I'm really stuck on this one.
Help would be appreciated. 
I'm given this matrix 
$$\left[
\begin{array}{ccc|c}
2&3&0&9\\0&1&\lambda+6&4\\
0&0&\lambda^2-5\lambda+6&9-3\lambda
\end{array}\right],
$$
and I need to find for which  λ ∈ ℝ this matrix has 


*

*one solution

*no solutions

*infinite solutions


Use the determinant formula for the first three columns of the matrix (that compose an upper triangular matrix, so you just need to get the product of the diagonal elements).
There is one solution if and only if the determinant you got is not equal to zero (Determinant theorem).
Then, check the values for which the determinant is equal to 0 and write the new matrices using that values.
For every new matrix, you have to study how many dependent/independent rows or columns you have (you can use Gauss algorithm, too).
Knowing that, you can have a conclusion using the Rouché-Capelli theorem.
