math111_logo Existence


Exercise Which of the following systems are consistent?

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

Answer

Exercise For what choices of the parameters are the following systems consistent?

x + 2y = a
3x + 4y = b
x + y = a
2x + 2y = b
3x + 3y = c
4x + 4y = d
x1 + x2 + 2x3 + x4 = 1
x1   + 2x3   = 0
2x1 + 2x2 + 3x3   = h
  x2 + x3 + 3x4 = h
    u + 2v = 1
- 2x + 2y - u + 4v = h
x - y + u - v = 1
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 Which of the following systems are consistent for any right side?

2x1   + 2x3 + x4 = b1
x1 + x2 - x3   = b2
  x2 - 2x3 + x4 = b3
x1 + 2x2 + x3 = b1
2x1 + x2 + x3 = b2
  3x2 + x3 = b3
3x1   + x3 = b4
x1 - x2 + x3   + 2x5 = b1
2x1 - 2x2   + 2x4 + 2x5 = b2
- x1 + x2 + 2x3 - 3x4 + x5 = b3
- 2x1 + 2x2 + x3 - 3x4 - x5 = b4

Answer

Exercise For what choices of the parameters are the following systems consistent for any right side?

x1 + x2 - x3 = b1
x1   - ax3 = b2
x1 + ax2   = b3
x1 + x2 + x3 + x4 = b1
x1     + ax4 = b2
  x2 + ax3   = b3
  v + w = b1
u   + aw = b2
u + v   = b3
  av - w = b4

Answer

Exercise List all shapes of the row echelon forms of the augmented matrices of consistent systems of 2 equations and 2 variables. What about systems of other sizes?

Answer

Exercise List all shapes of the row echelon forms of 2 by 2 matrices A so that Ax = b has solutions for any b. What about matrices of other sizes?

Answer

Exercise Without any computation, determine the consistency of the systems.

2x1   + 2x3 + x4 = 2
x1 + x2 - x3   = -1
  x2 - 2x3 + x4 = -2
x1 + 2x2 = 1
2x1 + 4x2 = 2
x1 + 2x2 = a
- x1 - 2x2 = b
x1 + x2 + x3 = 1
x1 + 2x2 + 3x3 = 2
2x1 + 3x2 + 4x3 = 0

Answer

Exercise Show that the system Ax = 0, with the zero vector on the right side, is always consistent.

Answer