Cholesky factorization

Task number: 2727

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

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

Difficulty level: Easy task (using definitions and simple reasoning)
Routine calculation training
Cs translation
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