Problems:
Systems of Linear Equations : Row Operations and Solutions
1. Find the augmented matrix for each system of linear equations:
a. 5x1 + 7x2 + 8x3 = 3
-2x1 + 4x2 + 9x3 = 3
3x1 - 6x2 + x3 = 1
b. 4x1 + x2 - 7x3 = 6
5x1 + 7x2 + 2x3 = 3
5x1 + 2x2 + 5x3 = 7
c. 3x1 - 2x2 + 2x3 = 7
5x1 + 7x2 + 3x3 = 3
-5x1 + 6x2 - 8x3 = -5
2. Using elementary row operations reduce each of the augmented matrices from Problem 1 to reduced echelon form
3. Using the information from Problem 2, what are the solutions to the system of linear equations
4. Indicate whether the following statements are True or False
a. Elementary row operations on an augmented matrix never change the solution set of the associated linear system. (Give a brief justification for your answer.)
b. Two matrices are row equivalent if they have the same number of rows. (Give a brief justification for your answer.)
c. An inconsistent system has more than one solution. (Give a brief explanation for your answer.)
d. Two linear systems are equivalent if they have the same solution set. (Give a brief explanation for your answer.)
5. Determine if the homogenous linear systems below have non-trivial solutions.
a. 2x1 - 5x2 + 8x3 = 0
-2x1 - 7x2 + x3 = 0
4x1 + 2x2 + 7x3 = 0
b. x1 - 3x2 + 7x3 = 0
-2x1 + x2 - 4x3 = 0
x1 + 2x2 + 9x3 = 0