### 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

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

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

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

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?

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?