Algebra Quiz 1: Reduced Row Echelon Form and Linear Systems
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Questions and Answers

If the matrix

0 0 2
1 + √6 3 6 - √2
0 1 1 + √6

is in reduced row echelon form (RREF), then l is equal to:

  • 1 - √6 (correct)
  • None
  • √6
  • -√6
  • 1 + √6

    • Consider the linear system

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

    If x = -1, y = 2, and z = r is the solution of this system, where a ∈ ℝ and r ∈ ℝ, then a is equal to:

  • 5 (correct)
  • -4
  • 4
  • None
  • -5

    • The linear system

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

    has:

  • no solution (correct)
  • None
  • a unique solution
  • the trivial solution
  • infinitely many solutions

    • Consider the linear system

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

    If z = t, t ∈ ℝ, then y is equal to:

    <p>y is equal to: -4 + 10*t</p> Signup and view all the answers

    Consider the linear system above. If x is equal to:

    <p>x is equal to: 13 + 6*t</p> Signup and view all the answers

    Study Notes

    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.

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    Description

    This 20-minute quiz covers matrix operations in reduced row echelon form and solving linear systems. It includes multiple-choice questions to test your understanding of algebra concepts.

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