Solving Systems of Equations with Matrices on the TI-84

Algebra 2 Grades High School 2:29 Video
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Lesson Description

Learn how to use the TI-84 calculator to solve systems of equations using inverse matrices. This lesson covers rewriting systems as matrix equations and utilizing the inverse matrix method.

Video Resource

Using Matrices to Solve Systems of Equations on Ti84 Calculator

Mario's Math Tutoring

Duration: 2:29
Watch on YouTube

Key Concepts

  • Matrix Representation of Systems of Equations
  • Inverse Matrices
  • TI-84 Calculator Matrix Operations

Learning Objectives

  • Students will be able to rewrite a system of linear equations in matrix form.
  • Students will be able to use the TI-84 calculator to find the inverse of a matrix.
  • Students will be able to solve a system of equations using inverse matrices on the TI-84 calculator.

Educator Instructions

  • Introduction (5 mins)
    Briefly review systems of equations and methods of solving them (substitution, elimination). Introduce the concept of using matrices as an alternative method. Show the video.
  • Writing Systems as Matrix Equations (10 mins)
    Explain how to represent a system of equations in the form AX = B, where A is the coefficient matrix, X is the variable matrix, and B is the solution matrix. Provide examples and have students practice converting systems of equations into matrix form.
  • Understanding Inverse Matrices (10 mins)
    Explain the concept of an inverse matrix and its properties (A * A⁻¹ = I, where I is the identity matrix). Discuss how multiplying both sides of the matrix equation AX = B by A⁻¹ solves for X. Avoid manual calculation of inverses at this stage, focusing on the concept.
  • TI-84 Calculator Demonstration (15 mins)
    Follow along with the video, demonstrating how to input matrices A and B into the TI-84 calculator. Show how to calculate A⁻¹ * B to solve for X. Emphasize the importance of the order of multiplication (A⁻¹ * B, not B * A⁻¹).
  • Practice Problems (15 mins)
    Provide students with several systems of equations to solve using the matrix method on their TI-84 calculators. Circulate to provide assistance and answer questions.
  • Wrap-up (5 mins)
    Summarize the steps involved in solving systems of equations using matrices on the TI-84. Address any remaining questions.

Interactive Exercises

  • Matrix Equation Transformation
    Given a system of equations, students work in pairs to transform it into a matrix equation (AX = B).
  • Calculator Relay
    Divide the class into teams. Each team receives a system of equations. Team members take turns entering the matrices and performing the calculations on the TI-84 to solve the system.

Discussion Questions

  • Why is it important to multiply by the inverse matrix on the correct side (left or right)?
  • What are the advantages of using matrices to solve systems of equations compared to other methods (substitution, elimination)?
  • How can you tell if a system of equations has no solution or infinitely many solutions when using the matrix method?
  • What are the limitations of using a graphing calculator to solve a system of equations?

Skills Developed

  • Matrix Algebra
  • Problem-Solving
  • Calculator Proficiency
  • Abstract Reasoning

Multiple Choice Questions

Question 1:

Which matrix represents the coefficients in the following system of equations? x + 2y = 5; 3x - y = 1

Correct Answer: [1, 2; 3, -1]

Question 2:

In the matrix equation AX = B, what does 'X' represent?

Correct Answer: Variable Matrix

Question 3:

What operation do you perform on the matrices A and B to solve for X in the equation AX = B?

Correct Answer: A⁻¹ * B

Question 4:

What is the first step to do on the TI-84 calculator to input a matrix?

Correct Answer: Press '2nd' then 'MATRIX'

Question 5:

After entering the matrix values, what step must you take to go back to the homescreen?

Correct Answer: Press '2nd' then 'QUIT'

Question 6:

The inverse of a matrix A, denoted as A⁻¹, has the property that A * A⁻¹ equals:

Correct Answer: The identity matrix

Question 7:

When solving AX = B for X, which expression correctly isolates X?

Correct Answer: X = A⁻¹ * B

Question 8:

What does the TI-84 calculator use to represent the inverse of a matrix?

Correct Answer: x⁻¹

Question 9:

What is the identity matrix similar to in normal arithmetic?

Correct Answer: One

Question 10:

What key must you press to get the matrix functions on a TI-84 calculator?

Correct Answer: 2nd

Fill in the Blank Questions

Question 1:

A system of equations can be represented in matrix form as AX = ____.

Correct Answer: B

Question 2:

The matrix containing the coefficients of the variables is called the ______ matrix.

Correct Answer: coefficient

Question 3:

To solve for the variable matrix X, we multiply both sides of AX = B by the ______ of matrix A.

Correct Answer: inverse

Question 4:

On the TI-84 calculator, the matrix menu is accessed by pressing '2nd' and then the ______ key.

Correct Answer: MATRIX

Question 5:

The solution matrix contains the ______ values of the system of equations.

Correct Answer: constant

Question 6:

When multiplying matrices, the order of multiplication matters because matrix multiplication is not ______.

Correct Answer: commutative

Question 7:

The inverse of matrix A is denoted as ______.

Correct Answer: A⁻¹

Question 8:

The dimensions of a matrix are given as rows by ______.

Correct Answer: columns

Question 9:

The result of multiplying a matrix by its inverse is the ______ matrix.

Correct Answer: identity

Question 10:

On the TI-84, to calculate the inverse of matrix A, you enter A and then press the ______ key.

Correct Answer: x⁻¹

Educational Standards

A2.A.1.7:Represent and evaluate mathematical models using systems of linear equations with a maximum of three variables. Graphing calculators or other appropriate technology may be used. A2.N.2:Extend the understanding of numbers and operations to matrices. A2.N.2.1:Use matrices to organize and represent data. Identify the order (dimension) of a matrix.

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