Homework 6: Ax=b

  1. Solve the following linear system using Gaussian elimination. Show your work.

    \begin{pmatrix} 4 & 2 & 0 \\ 0 & 2 & 1 \\ 1 & 0 & 1 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} = \begin{pmatrix} 1 \\ 2 \\ 0 \end{pmatrix}

  2. Solve the same system \mod 3. Show your work.
  3. Given an efficient subroutine that takes integer A,b as input and finds a rational x such that Ax=b, give a simple, efficient algorithm to find a solution to Ax=b  \mod p for any prime p that does not divide \det(A).
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