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Solutions from Matrices in RRE Form

Mathematical Programming, Sean Forman

Spring 2005

Each of the following matrices is in Reduced Row-Echelon form. For each matrix in RRE do the following.

• Circle the leftmost ones.

• State whether the systen has no solution (a leftmost one in the last column).

• List the basic variables (variables that have leftmost ones).

• List the non-basic variables (variables without leftmost ones).

• Assign the non-basic variables an arbitrary value ($t$ or $t_1, t_2,\ldots$).

• Solve for the basic variables (may be constants or may be in terms of $t$ or $t_1, t_2,\ldots$). What would these be if we assumed the non-basic variables were all zero.

1. $\left( \begin{array}{r r r r} 1 & 0 & 0 & 3 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & -4 \end{array}\right)$
   
   
   

2. $\left( \begin{array}{r r r r} 1 & 1 & 0 & 4 \\ 0 & 0 & 1 & 6 \\ 0 & 0 & 0 & 0 \end{array}\right)$
   
   
   

3. $\left( \begin{array}{r r r r} 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right)$
   
   
   

4. $\left( \begin{array}{r r r r r} 1 & 0 & 0 & 2 & -5 \\ 0 & 1 & 0 & 2 & 4\\ 0 & 0 & 1 & 1 & 10\\ 0 & 0 & 0 & 0 & 0 \end{array}\right)$
   
   
   

5. $\left( \begin{array}{r r r r} 1 & 0 & 1 & 6 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right)$
   
   
   

6. $\left( \begin{array}{r r r r r} 1 & 0 & 1 & 2 & -5 \\ 0 & 1 & 0 & 2 & 4\\ 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 \end{array}\right)$
   
   
   

7. $\left( \begin{array}{r r r r r} 1 & 0 & 0 & 0 & -5 \\ 0 & 1 & 0 & 0 & 4\\ 0 & 0 & 1 & 0 & -8\\ 0 & 0 & 0 & 1 & 10 \end{array}\right)$
   
   
   

8. $\left( \begin{array}{r r r r r} 1 & 0 & 0 & 2 & -5 \\ 0 & 1 & 0 & 2 & 4\\ 0 & 0 & 1 & 0 & 12\\ 0 & 0 & 0 & 0 & 0 \end{array}\right)$
   
   
   

9. $\left( \begin{array}{r r r r r} 1 & 0 & 1 & 0 & -5 \\ 0 & 1 & -8 & 0 & 4\\ 0 & 0 & 0 & 1 & -4\\ 0 & 0 & 0 & 0 & 0 \end{array}\right)$