Worksheet 3

1. Use Gaussian Elimination to convert the following system into row echelon form \(x_1 + 2x_2 - x_5 + x_7 = 4 \\ x_2 - x_3 + x_4 - 3x_5 + x_6 = 0 \\ -3x_1 + x_2 + x_3 + x_4 + x_5 + x_7 = 1 \\ x_3 - x_6 - 2x_7 = 4 \\ x_1 - x_2 + x_3 + x_4 + x_5 - x_6 + x_7 = 0\) Use this to find the general solution to the system.

2. Use Gaussian Elimination to convert the following system into reduced row echelon form. \(2x_1 - x_2 + x_3 + x_4 = -1 \\ x_2 - x_3 + x_4 = 1 \\ x_1 + x_2 + x_3 + x_4 = 1\) Use this to find the general solution to the system.

3. We say two matrices $A$ and $A^\prime$ are row-equivalent if we can obtain $A$ from $A^\prime$ by performing some number of row exchanges, row scaling, and by adding scalar multiples of one row to another. We are allowed to these operations in any order with any choices of scalars and rows.

   Gaussian Elimination shows that any matrix is row-equivalent to one in reduced row echelon form because it gives an algorithm for producing it.

   Can you have two matrices $A$ and $A^\prime$ that are both in row echelon form and are row-equivalent but are not equal?

   Can you have two matrices $A$ and $A^\prime$ that are both in row echelon form and are row-equivalent but are not equal?