Question: I looked up the solution to a math problem in the solution manual, and I'm having trouble understanding how the solution works.

In this problem the lagrange method is used, the part I'm having trouble understanding is how they solved this linear equation:

[![enter image description here](https://i.stack.imgur.com/JXkyL.png)](https://i.stack.imgur.com/JXkyL.png)

Which gives this system of equations:

[![enter image description here](https://i.stack.imgur.com/NyD8Z.png)](https://i.stack.imgur.com/NyD8Z.png)

**The solution and the part I don't understand that they showed in the solution manual:**

*Clearly this has the solution (x, y) = (0, 0) for there to be other solutions the following has to be true:*

[![enter image description here](https://i.stack.imgur.com/cycUF.png)](https://i.stack.imgur.com/cycUF.png)

*Which is fulfilled for λ = 1/3 and λ = -1*

What I don't understand is how does this solution work? Why is the determinant of that matrix 0 when there are other solutions?

Answer: 
The matrix equation $A\vec x=\vec 0$ has solutions other than $\vec x=\vec 0$ exactly when $\det(A)=0$. This is one of the many parts of the [invertible matrix theorem](http://mathworld.wolfram.com/InvertibleMatrixTheorem.html). Your set of equations can be seen as such a matrix equation with $\vec x=\left(\begin{smallmatrix}x\\y\end{smallmatrix}\right)$, and $A$ as the matrix they have there.

You can, of course, just solve your set of equations and come to the same conclusion the "hard" way (it isn't that hard, and it's essentially what proves that part of the IMT). It's just that invoking the IMT makes the solution a lot shorter, and more readable, assuming you know why it's true.
λ = 1/3 and λ = -1 are just eigen values for the homogeneous system. They are the ones that give rise to non-zero solutions.