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Homework 3 - Solvability and LU factorizations

Due 2021-09-14.

  1. Find a matrix $B$ whose null space equals the range of the matrix \(A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \\ 7 & 8 \\ 9 & 10 \end{pmatrix}\)

  2. Determine if the following matrix $A$ has an LU factorization. If not, can you find a permutation matrix $P$ such that $PA$ has an LU factorization? \(A = \begin{pmatrix} 0 & 1 & 2 \\ -2 & 0 & 1 \\ 1 & 1 & 1 \end{pmatrix}\)

  3. Let $L_1$ and $L_2$ be two lower triangular matrices:
    • Show that $L_1 + L_2$ is lower triangular.
    • If both $L_1$ and $L_2$ are square, show that $L_1 L_2$ is also lower triangular.
  4. A matrix $A$ is called orthogonal if $A^{-1} = A^T$. In other words, $A$ is an orthogonal matrix if its inverse is equal to its transpose. Show that any permutation matrix is orthogonal.

  5. Show that the product of two orthogonal matrices is also orthogonal.