Algebra 2 Section 13 - Mrs. Hart's Math



College Algebra Reduced Row Echelon Form (RREF) Name__________________

|From the augmented matrix given, determine whether the system is consistent and independent, consistent and dependent, or |

|inconsistent. Give solution if possible, and if dependent, give family of solutions. |

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|[pic] 2. [pic] 3. [pic] |

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|Solve each system of equations by using the augmented matrix method. Determine whether the system is consistent and independent, |

|consistent and dependent, or inconsistent. If dependent, give family of solutions. |

|4. [pic] |5. [pic] |6. [pic] |

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|7. [pic] |8. [pic] |9. [pic] |

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|10. [pic] |11. [pic] |12. [pic] |

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|13. [pic] |14. [pic] |15. [pic] |

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|Write the system of equations represented by each matrix. Do not solve. |

|16. [pic] | | |

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Solving Systems of Equations with Reduced Row Echelon Form command

Vocabulary Augmented Matrix: the coefficient matrix with added column(s)

Example: Solve 2x + 3y – z = 7

-3x – y + 2z = -4

x + 4y + 2z = 6

Solution: Write augmented matrix with coefficient matrix augmented with constant column

[pic] Enter this 3 x 4 matrix into calculator and call it A.

On main screen, go to MATRIX, then MATH, then choose rref(

Input the matrix you are going to be taking the rref of, namely, rref([A]) and hit enter.

Out pops the reduced row echelon form (RREF) of the matrix, where you can read the answers.

This problem results in a RREF of [pic] and we can reinterpret into math equations, that is, 1x + 0y + 0z = 0 or x = 0

0x + 1y + 0z = 2 or y = 2

0x + 0y + 1z = -1 or z = -1 and our solution is (0, 2, -1) which we could check in.

RREF stands for Reduced Row Echelon Form which means you could get a matrix that looks like where the left 3 columns is I, the identity matrix, and the last .column is your solution.

RREF can also be used for inconsistent systems and consistent dependent systems, unlike using Cramer’s Rule or using inverse matrices to solve the equation where you just get error messages.

|If result looks like |If result looks like |If result looks like |

|[pic] |[pic] |[pic] |

|and we can recognize |and we can recognize |and we can recognize |

|that we have Identity |that we have DO NOT have Identity on left, we |that we DO NOT have Identity on left, we see |

|on left, we have solutions |see that bottom row shows 0 = 2 which is false |that bottom row shows 0 = 0 which is true! |

|on right | | |

| |Inconsistent |Write equations from first two rows: |

|Consistent Independent |No solution |1x + 0y + 0z = 5 or x = 5 and |

|(5, 3, 2) |would be classification and |y + z = 3 |

|would be classification and |solution |solve y in terms of z or z in terms of y: y =|

|solution | |3 – z |

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| | |Consistent Dependent |

| | |(5, 3 – z, z) is one way to give family of |

| | |solutions. |

| | |would be classification and |

| | |solutions |

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