math111_logo 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

Answer

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

Answer

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?

Answer

Exercise Let A be a 3 by 3 matrix. Prove that the following are equivalent

What about the general case that A is an n by n matrix?

Answer