Cholesky factorization

Decide whether the following matrix is positive definite by Gauss elimination and by use of determinants. If so, find its Cholesky factorization.

• Variant 1

$$\begin{pmatrix} 2 & 1 & 0 \\ 1 & 2 & 0 \\ 0 & 0 & 4 \\ \end{pmatrix}$$

• Variant 2

$$\begin{pmatrix} 1 & 2 & 1 & 0 \\ 2 & 8 & 4 & 2\\ 1 & 4 & 11 & 1 \\ 0 & 2 & 1 & 2 \\ \end{pmatrix}$$

• Variant 3

$$\begin{pmatrix} 4 & 2 & 2 & 0 \\ 2 & 2 & 4 & 1\\ 2 & 4 & 10 & 3 \\ 0 & 1 & 3 & 6 \\ \end{pmatrix}$$

• Variant 4

$$\begin{pmatrix} 4 & 0 & 0 & 2 \\ 0 & 1 & 3 & 0\\ 0 & 3 & 10 & 1 \\ 2 & 0 & 1 & 3 \end{pmatrix}$$

• Variant 5

$$\begin{pmatrix} 1 & -2 & 1 & 1 & -1\\ -2 & 8 & -2 & -4 & 8\\ 1 & -2 & 2 & 3 & -1\\ 1 & -4 & 3 & 15 & -1\\ -1 & 8 & -1 & -1 & 15 \end{pmatrix}$$