I am trying to understand the exact form of the Cholesky algorithm used in the following exercise:
Question: Find all values of $\rho$ such that the following matrix is symmetric positive semidefinite: $$ \Omega = \begin{bmatrix}1 & 0.6 & -0.3\\0.6 & 1 & \rho \\ -0.3 & \rho & 1\end{bmatrix} $$
Answer: Apply the first step of the Cholesky algorithm to obtain the following 2x2 matrix, note its (1,1) entry is $\ge 0$ and therefore stipulate its determinant be $\ge 0$: $$ M=\begin{bmatrix}1 & \rho \\\rho & 1 \end{bmatrix} - \begin{bmatrix}0.6 \\ -0.3 \end{bmatrix} \begin{bmatrix} 0.6 & - 0.3\end{bmatrix} = \begin{bmatrix}0.64 & \rho + 0.18 \\\rho + 0.18 & 0.91 \end{bmatrix}$$ $$ M(1,1) = 0.64 \ge 0$$ $$ \det M = 0.5824 - (\rho+0.18)^2 \ge 0$$
How does the first step of this algorithm relate to its description in Wikipedia (say): https://en.wikipedia.org/wiki/Cholesky_decomposition#The_Cholesky_algorithm