This document discusses representing and solving systems of linear equations using matrices. It defines what a matrix is and how to identify matrix elements. A system of equations can be represented by a matrix with each row representing an equation and each column representing a variable, except the last column which holds the constants. To solve the system, the matrix is row reduced into reduced row echelon form through operations like row switching, scalar multiplication, and row addition. The solutions can then be read from the reduced matrix. Graphing calculators can also use the rref function to row reduce a matrix representing a system of equations and directly give the solutions.

1. 3-6 Solving Systems
Using Matrices
Algebra II Unit 3 Linear Systems
© Tentinger

2. 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

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

4. 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.

5. 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?

6. 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

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

8. What linear system of equations
does the matrix represent?

9. Solving Systems
• To solve a system of equations using matrices it should look
like this when finished row reducing:
• where
x=a
y=b
z=c

10. Row operations:
• Switch any two rows:
Becomes
• Multiply a row by a constant:
Becomes
• Add one row to another:
Becomes
• Combine any of these steps
• Matrices that represent the solution of a system are in
Reduced Row Echelon Form

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

12. 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 rref
function.
• 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?

13. Homework
• Pg 179-180
• #8-14, 18 – 20, 24-26, 30, 34, 38
• 15 problems