Solving system of 3 equations with matrix

I am studying about Larange Multiplier on MIT OCW and I stumbled through an algebra step that I'm not quite sure about.

These are the pics from the lecture.

The equations - pic 1

Matrix transformation - pic 2

Please help me explain how he translates the system of equations into matrices form and then determine the solution. Plus, why is the determinant of matrix M in this case equals to C would give the solution?

What should I take in order to understand these concept more in depth? (I'm guessing linear algebra but I'm not too sure).

Thank you very much!

$\begin{bmatrix} 2 & -\lambda\\ \lambda & -2 \end{bmatrix} \cdot \begin{bmatrix}x\\y\end{bmatrix} = \begin{bmatrix}2x-\lambda y\\ \lambda y -2x\end{bmatrix} = \begin{bmatrix}0\\0\end{bmatrix}$
$2x-\lambda y = 0$ and $\lambda y -2x = 0$
You can always bring a linear system of equations into matrix form by letting the matrix $M$ be the matrix of coefficients, i.e. $M_{i,j}$ is the coefficient in the i-th equation of the j-th variable.