### Uniqueness

Exercise Which of the following consistent systems have a unique solution (compare with this exercise)?

 2x1 + 2x3 + x4 = 2 x1 + x2 - x3 = -1 x2 - 2x3 + x4 = -2
 x1 + 2x2 + x3 = 1 2x1 + x2 + x3 = 1 3x2 + x3 = 1 3x1 + x3 = 1
 x1 - x2 + x3 + 2x5 = 1 2x1 - 2x2 + 2x4 + 2x5 = 0 - x1 + x2 + 2x3 - 3x4 + x5 = 2 - 2x1 + 2x2 + x3 - 3x4 - x5 = 1
 x1 - x2 + x3 + 2x5 = 3 2x1 - 2x2 + 2x4 + 2x5 = 2 - x1 + x2 + 2x3 - 3x4 + x5 = 3 - 2x1 + 2x2 + x3 - 3x4 - x5 = 0
 x1 - x2 = 1 - 2x1 + 3x2 + x3 = 2 x1 + x3 = 5 x1 + 2x2 + 3x3 = 13 2x1 - 2x2 + 4x3 = 5

Exercise For what choices of the parameters do the following systems have a unique solution (compare with this exercise)?

 x + 2y = a 3x + 4y = b
 x1 + x2 - x3 = -2 x1 - ax3 = -1 x1 + ax2 = -1
 x1 + x2 + x3 + x4 = 1 x1 + ax4 = 1 x2 + ax3 = 1
 v + w = a u + aw = 1 u + v = 0 av - w = 1

Exercise List all shapes of the row echelon forms of the augmented matrices of consistent systems of 2 equations in 2 variables with a unique solution. What about systems of other sizes?