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Warm-Up. Solve the following system of equations:. Reduced Row Echelon Form (RREF). Learning Targets. Possible solutions for a system The differences between RREF and Inverse Multiplication Using Reduced Row Echelon Form to solve systems. Quick Recap.

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Warm-Up

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Warm up

Warm-Up

  • Solve the following system of equations:


Reduced row echelon form rref

Reduced Row Echelon Form (RREF)


Learning targets

Learning Targets

  • Possible solutions for a system

  • The differences between RREF and Inverse Multiplication

  • Using Reduced Row Echelon Form to solve systems


Quick recap

Quick Recap

  • In order to setup a matrix our data must be:

    • Setup in the Standard Form

    • All similar variables must be in the same order

    • Any missing variables in the equations must be represented with a zero

  • Our matrix system using inverse multiplication has three matrices:

    • Coefficient, Variable and Constant


Recap cont

Recap Cont.

  • Matrices cannot divide one another

  • Multiplying the coefficient matrix by its inverse isolates the variable matrix

  • Multiplying the constant matrix by our inverse as well will solve for the variables


How many types of solutions can we have

How many types of solutions can we have?

  • Pause and Ponder, SILENTLY!

  • One Solution

  • No Solution

  • Infinite Solutions


Example

Example

Solve using Inverse Multiplication:

What Does it Mean?!?


Example1

Example

Solve using Inverse Multiplication:

AGAIN!?!


Problems with inverse multiplication

Problems with Inverse Multiplication

  • Inverse multiplication only produces a solution when there is only one solution.

  • If we have no solution or infinite solutions then we will get an ERROR

  • So how do we know if it is infinite solutions or no solution?


Warm up

RREF

  • Reduced Row Echelon Form

    • This form allows us to consolidate the coefficient and constant matrices into one matrix

    • We can then perform row operations that will clearly state the exact answer


How to set it up

How to set it up:

Old Way:

Coefficient

Variable

Constant


How to set it up1

How to set it up:

***Notice there is

no more variable

matrix and we add

the constant as an

additional column.

RREF

Coefficient

Constant


Now to perform the math

Now to perform the math…

  • In our calculators we can go under the matrix screen and select RREF

  • We can then choose the matrix to perform this operation on.


Solution

Solution

  • The solution will be in a matrix that is the same dimensions.

  • We can then read the results as variables and their solutions.

X-Variable

Y-Variable

Z-Variable


No solution

No Solution:

Enter as a 3X4 matrix

Last row:

0 0 0 1

No Solution.


Infinite solutions

Infinite Solutions:

Enter as a 3X4 matrix

Last row:

0 0 0 0

Infinitely Solutions


You try using rref

You Try using RREF:


Website to visualize the solutions

Website to Visualize the Solutions

  • http://www.cpm.org/flash/technology/3dsystems.swf


For the weekend

For the weekend…

  • Worksheet


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