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**Solving systems using matrices**

4.4 Solving systems using matrices

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**“A” Matrix “The Matrix” A matrix is a rectangular array of numbers.**

“The Matrix” is a movie with Keanu Reeves. “The Matrix”

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**Example of a matrix Columns Rows Element**

Note: A Square matrix has the same # of rows and columns

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**Writing an Augmented Matrix**

Linear Equations 1: Linear Equations 2: Augmented Matrix Note these are Standard Form

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**Writing an Augmented Matrix**

Linear Equations 1: Linear Equations 2: Augmented Matrix Note these are Standard Form EX. 1

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**Writing an Augmented Matrix**

Linear Equations 1: Linear Equations 2: Augmented Matrix Write in Standard Form!!! EX. 2

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Row Transformations All numbers in a row may be multiplied or divided by any nonzero real number. You can replace rows by adding them to other rows and placing the sum in the row.

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**Transformations Example 1**

All numbers in a row may be multiplied or divided by any nonzero real number. Multiply R1 by -2 =

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**Transformations Example 2**

All numbers in a row may be multiplied or divided by any nonzero real number. Divide R2 by 3 =

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Example 2 ANSWER

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**Transformations Example 3**

All numbers in a row may be multiplied or divided by any nonzero real number. Multiply R1 by 2 and multiply R2 by -4 =

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Example 3 ANSWER

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**Transformations Example 4**

You can replace rows by adding them to other rows and placing the sum in the row. Replace R1 with R1+R2 =

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**Transformations Example 5**

You can replace rows by adding them to other rows and placing the sum in the row. Replace R2 with R1-R2 =

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Example 5 ANSWER

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**Transformations Example 6**

Replace R1 with : -2R1 + R2 =

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Example 6 ANSWER Note: R2 does not change!!!!

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**Transformations Example 7**

Replace R2 with : -1/2R2 – R1 =

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Example 7 ANSWER Note: R1 does not change!!!!

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**Triangular form The 1’s and the 0 in these locations**

a, p, and q are just constants

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**Use row transformation to get a matrix in triangular form**

1.Work in column 1 to get the one. 2. Get the zero in column 1. 3. Get the 1 in column 2. 1st 2nd 3rd

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**Triangular form Example 1**

Write the matrix in Triangular form =

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**Example 1 Steps 1st : 1/6 R1 2nd : Replace R2 with 10R1 + R2**

3rd : -1/28 R2 Let’s Look at it !

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Example 1 ANSWER

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**Triangular form Example 2**

Write the Linear Equations in standard form. Write the Augmented Matrix. Get the matrix in Triangular Form. Write the matrix back into Standard form. Solve for x and y.

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Put in Standard form. 2. Write the Augmented Matrix

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**3. Try for Triangular Form.**

4. Back to Standard Form.

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**5. Solve for x and y. Looking here. Therefore ( 7/2 , -1)**

y = -1, now substitute into equation 1. x = 7/2 Therefore ( 7/2 , -1) is where the lines cross.

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**Make sure to review these notes!**

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