3-6 Solving Systems Using MatricesAlgebra II Unit 3 Linear Systems

© Tentinger

Essential Understanding and Objectives

• Essential Understanding: You can use a Matrix to represent and solve a system of equations without writing the variables.

• Objectives:

• Students will be able to:

• Represent a system of linear equations with a matrix

• Solve a system of equations using matrices

Iowa Core Curriculum

• Algebra

• A.REI.8 (+) Represent a system of linear equations as a single matrix equation in a vector variable.

Matrix

• Matrix: a rectangular array of numbers. Displayed with brackets and the dimensions are the number of rows by the number of columns.

• Dimensions: 2x3

• Matrix Element: each number in the matrix. Represented by its row and column number as a subscript. For example in matrix A, a12 is the element in row 1 column 2 and is the fourth element.

Identify a Matrix Element.

• A =

• What is element a23 in matrix A?

• What is element a13 in Matrix A

• What is element a33 in matrix A?

• What is element a24 in matrix A?

Representing a system of equations with a matrix.• Each row represents an equation

• Each column represents a variable, except the last column

• The last column shows the constants to the right of the equal signs.

• You only write the constant with the variable in the matrix

• Draw a vertical bar to replace the equal signs

• System Matrix

How can you represent the system of equations with a matrix?

What linear system of equations does the matrix represent?

Solving Systems

• To solve a system of equations using matrices it should look like this when finished row reducing:

• where x = ay = bz = c

Row operations:• Switch any two rows:

• Multiply a row by a constant:

• Add one row to another:

• Combine any of these steps

• Matrices that represent the solution of a system are in Reduced Row Echelon Form

Becomes

Becomes

Becomes

Solving a system using a matrix• What is the solution of the system?

On the Calculator

• Matrices that represent the solution of a system are in Reduced Row Echelon Form

• Graphing Calculators will do the work for you using the rreffunction.

• On the Calculator

• Enter in the system into the matrix

• Apply the rref() function to the matrix

• List the solutions

• What is the solution of the system of equations?

Homework

• Pg 179-180

• #8-14, 18 – 20, 24-26, 30, 34, 38

• 15 problems