Cholesky factorization of a 2×2 matrix

Setup.
$$LL^T = \begin{pmatrix} a & 0 \\ b & c \end{pmatrix}\begin{pmatrix} a & b \\ 0 & c \end{pmatrix} = \begin{pmatrix} a^2 & ab \\ ab & b^2+c^2 \end{pmatrix}.$$

Identification. $a^2 = 4$, so $a = 2$ (positive). $ab = 2$, so $b = 1$. $b^2+c^2 = 5$, so $c^2 = 4$, $c = 2$ (positive).

Conclusion.
$$L = \begin{pmatrix} 2 & 0 \\ 1 & 2 \end{pmatrix}.$$

Verification.
$$LL^T = \begin{pmatrix} 2 & 0 \\ 1 & 2 \end{pmatrix}\begin{pmatrix} 2 & 1 \\ 0 & 2 \end{pmatrix} = \begin{pmatrix} 4 & 2 \\ 2 & 5 \end{pmatrix}.$$