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 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. 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

[ 𝐴 ]× [𝑣 ]= [𝐵 ]

[𝑣 ]=[ 𝐴 ]−1× [𝐵 ]

How many types of solutions can we have?Pause and Ponder, SILENTLY!

One Solution

No Solution

Infinite Solutions

ExampleSolve using Inverse Multiplication:

What Does it Mean?!?

ExampleSolve using Inverse Multiplication:

AGAIN!?!

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?

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:

3 1 11 −2 3−2 3 −1

𝑥𝑦𝑧

101114

¿Old Way:

Coefficient Variable Constant

How to set it up:

3 1 11 −2 3−2 3 −1

101114

RREF

Coefficient Constant

***Notice there isno more variablematrix and we addthe constant as an additional column.

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 The solution will be in a matrix that is the same dimensions. We can then read the results as variables and their solutions.

X-VariableY-VariableZ-Variable

No Solution:Enter as a 3X4 matrix

Last row:

0 0 0 1No Solution.

Infinite Solutions:

Enter as a 3X4 matrix

Last row:

0 0 0 0Infinitely Solutions

You Try using RREF:

Website to Visualize the Solutions

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

For the weekend…Worksheet