Computational Problems

Gaussian Elimination

Problem 1:

Find the reduced row echelon form of the following matrices by hand:

  1. \(\begin{bmatrix} -1 & 1 \\ -1 & 0 \\ 0 & -1 \\ -1 & 2 \end{bmatrix}\)
  2. \(\begin{bmatrix} 1 & 2 & 0 \\ 1 & 3 & 3 \\ -1 & 0 & -1 \\ -3 & 0 & 0 \end{bmatrix}\)
  3. \(\begin{bmatrix} 0 & -3 & 2 & -2 \\ 0 & 2 & 2 & -2 \end{bmatrix}\)

Problem 2:

Solve the following system of equations by hand:

\[ \begin{aligned} x_2 + 5x_3 &= -4 \\ x_1 + 4x_2 + 3x_3 &= -2 \\ 2x_1 + 7x_2 + x_3 &= -2 \end{aligned} \]

Problem 3:

Solve the following system of equations. Use SymPy to find the rref.

\[ \begin{aligned} 6x_3 + 2x_4 - 4x_5 - 8x_6 &= 8 \\ 3x_3 + x_4 - 2x_5 - 4x_6 &= 4 \\ 2x_1 - 3x_2 + x_3 + 4x_4 - 7x_5 + x_6 &= 2 \\ 6x_1 - 9x_2 + 11x_4 - 19x_5 + 3x_6 &= 1 \end{aligned} \]