Algebra Quiz 1: Reduced Row Echelon Form and Linear Systems

Matrix in Reduced Row Echelon Form (RREF)

• The matrix

  0 0 2 
  1 + √6 3 6 - √2 
  0 1 1 + √6 
  

is in RREF, and l is equal to one of the options (a) 1 - √6, (b) √6, (c) 1 + √6, (d) -√6, or (e) None.

Linear System Solution

• The linear system

  x + y + z = *a*
  2x + 3y + 3z = 16
  5x + 2y + 3z = 11
  

has a solution x = -1, y = 2, and z = r, where a ∈ ℝ and r ∈ ℝ.

Value of a

• The value of a is one of the options (a) 5, (b) -5, (c) 4, (d) -4, or (e) None.

Linear System Properties

• The linear system

  x + 2y + 5z =  4
  4x + y - 15z = 15
  2x + 3y + 5z =  5 
  

has one of the following properties: (a) a unique solution, (b) infinitely many solutions, (c) no solution, (d) the trivial solution, or (e) None.

Linear System Solution with z = t

• The linear system

  x + y + 3z = 2
  4x + 5y + 2z = 4
  3x + y + 29z = 14
  

has a solution z = t, where t ∈ ℝ.

• The value of y is one of the options (a) -4 - 10t, (b) 10 + 4t, (c) -4 + 10t, (d) 10 - 4t, or (e) None.

Value of x

• The value of x is one of the options (a) 13 + 6t, (b) 6 - 13t, (c) 13 - 6t, (d) 6 + 13t, or (e) None.

Linear System with Four Variables

• The linear system

  x + 2y - z + 3w = -5
  2x + 5y + 2z + 7w = -3
  3x + 8y + 6z + 9w = -2
  -2x - 5y + z - 12w = -1
  

is a system of four variables.