Solving systems using matrices 4.4 Solving systems using matrices

“A” Matrix “The Matrix” A matrix is a rectangular array of numbers. “The Matrix” is a movie with Keanu Reeves. “The Matrix”

Example of a matrix Columns Rows Element Note: A Square matrix has the same # of rows and columns

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

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

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

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.

Transformations Example 1 All numbers in a row may be multiplied or divided by any nonzero real number. Multiply R1 by -2 =

Transformations Example 2 All numbers in a row may be multiplied or divided by any nonzero real number. Divide R2 by 3 =

Example 2 ANSWER

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 =

Example 3 ANSWER

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 =

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 =

Example 5 ANSWER

Transformations Example 6 Replace R1 with : -2R1 + R2 =

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

Transformations Example 7 Replace R2 with : -1/2R2 – R1 =

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

Triangular form The 1’s and the 0 in these locations a, p, and q are just constants

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

Triangular form Example 1 Write the matrix in Triangular form =

Example 1 Steps 1st : 1/6 R1 2nd : Replace R2 with 10R1 + R2 3rd : -1/28 R2 Let’s Look at it !

Example 1 ANSWER

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.

Put in Standard form. 2. Write the Augmented Matrix

3. Try for Triangular Form. 4. Back to Standard Form.

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.

Make sure to review these notes!