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Linear Algebra

Worksheet 6

  1. Does the matrix \(A = \begin{pmatrix} 2 & -1 & -1 \\ 1 & 1 & -3 \\ -2 & 1 & 2 \end{pmatrix}\) have an LU decomposition? If so, find it.

  2. Does the matrix \(A = \begin{pmatrix} 1 & 0 & -1 & 4 \\ 1 & -1 & -1 & 2 \\ -2 & 1 & 2 & 1 \end{pmatrix}\) have an LU decomposition? If so, find it.

  3. For any matrix in 1 or 2 that does not have an LU decomposition, can you find a permutation matrix $P$ so that $PA$ does?