Inverse & Identity MATRICES Last Updated: October 12, 2005.

Slides:

Advertisements

Similar presentations

4.5 2x2 Matrices, Determinants and Inverses

Advertisements

Section 13-5 : Inverses of Matrices Objectives: 1)Background – what are inverses and why find them. 2)Process for finding the inverse of a 2x2 matrix.

Identity and Inverse Matrices

What is entry A in the matrix multiplication: ( a) 1 (b) -2(c) 5 (d) 11(e) 13 (f) 0.

MATRICES Using matrices to solve Systems of Equations.

Finding the Inverse of a Matrix

Using Matrices to Solve a 3-Variable System

Table of Contents Matrices - Inverse of a 2  2 Matrix To find the inverse of a 2  2 matrix, use the following pattern. Let matrix A be given by... Then.

4.3 The Inverse of a Matrix Warm-up (IN)

Table of Contents Matrices - Inverse Matrix Definition The inverse of matrix A is a matrix A -1 such that... and Note that... 1) For A to have an inverse,

4.7 Identity and Inverse Matrices. What is an identity? In math the identity is the number you multiply by to have equivalent numbers. For multiplication.

EXAMPLE 2 Solve a matrix equation SOLUTION Begin by finding the inverse of A = Solve the matrix equation AX = B for the 2 × 2 matrix X. 2 –7 –1.

Table of Contents Matrices - Inverse of a 3  3 Matrix Recall that to find an inverse of a 2  2 matrix, apply the formula... The formula for the inverse.

Inverse Matrices (2 x 2) How to find the inverse of a 2x2 matrix.

Using Inverse Matrices Solving Systems. You can use the inverse of the coefficient matrix to find the solution. 3x + 2y = 7 4x - 5y = 11 Solve the system.

DETERMINANTS AND INVERSE MATRICES. I. Determinants of 2x2 A. Formula.

 Definition: An Identity Matrix is a square matrix that, when multiplied by another matrix, equals the same matrix.  Form: In the Identity Matrix the.

4.5, x 2 and 3 x 3 Matrices, Determinants, and Inverses Date: _____________.

Matrix Equations Step 1: Write the system as a matrix equation. A three-equation system is shown below.

3.5 Solution by Determinants. The Determinant of a Matrix The determinant of a matrix A is denoted by |A|. Determinants exist only for square matrices.

MATRICES Jeffrey Bivin Lake Zurich High School Last Updated: October 12, 2005.

MAT 2401 Linear Algebra 2.3 The Inverse of a Matrix

4.5 Solving Systems using Matrix Equations and Inverses.

Matrix Entry or element Rows, columns Dimensions Matrix Addition/Subtraction Scalar Multiplication.

4.4 & 4.5 Notes Remember: Identity Matrices: If the product of two matrices equal the identity matrix then they are inverses.

Inverses and Systems Section Warm – up:

2.5 Determinants and Multiplicative Inverses of Matrices Objectives: 1.Evaluate determinants. 2.Find inverses of matrices. 3.Solve systems of equations.

Chapter 2 Systems of Linear Equations and Matrices

Do Now: Evaluate: 3AB. Algebra II 3.7: Evaluate Determinants HW: p.207 (4-14 even) Test : Friday, 12/6.

Ch X 2 Matrices, Determinants, and Inverses.

What you will learn 1. What an identity matrix is

4.7 Identity and Inverse Matrices and Solving Systems of Equations Objectives: 1.Determine whether two matrices are inverses. 2.Find the inverse of a 2x2.

Identity What number is the multiplication identity for real numbers? For matrices, n x n--square matrices, has 1’s on main diagonal and zeros elsewhere.

13.6 MATRIX SOLUTION OF A LINEAR SYSTEM.  Examine the matrix equation below.  How would you solve for X?  In order to solve this type of equation,

 1 is the multiplicative identify for real #’s : 1· a=a and a· 1 = a  For matrices n X n, the identity matrix has 1’s on its main diagonals and 0’s.

Chapter 4 Section 4: Inverse and Identity Matrices 1.

Advanced Trig Exam Review Day Three: Matrices. Solving Systems of Equations.

Inverse and Identity Matrices Can only be used for square matrices. (2x2, 3x3, etc.)

Have we ever seen this phenomenon before? Let’s do some quick multiplication…

2 x 2 Matrices, Determinants, and Inverses.  Definition 1: A square matrix is a matrix with the same number of columns and rows.  Definition 2: For.

2x2 Matrices, Determinants and Inverses

6.2/6.3: Properties of Parallelograms MCE Can you figure out the puzzle below??? Three Blind Mice.

2.2 The Inverse of a Matrix. Example: REVIEW Invertible (Nonsingular)

Assignment Questions?? Pg all, 23-26, 35, 46, 48 Handout Questions?

Chapter 4 Section 5 and 6 Finding and Using Inverses Algebra 2 Notes February 26, 2009.

Notes Over 4.4 Finding the Inverse of 2 x 2 Matrix.

Use Inverse Matrices to Solve Linear Systems Objectives 1.To find the inverse of a square matrix 2.To solve a matrix equation using inverses 3.To solve.

3.8B Solving Systems using Matrix Equations and Inverses.

13.3 Product of a Scalar and a Matrix.  In matrix algebra, a real number is often called a.  To multiply a matrix by a scalar, you multiply each entry.

EXAMPLE FORMULA DEFINITION 1.

Using Matrices to Solve a 3-Variable System

College Algebra Chapter 6 Matrices and Determinants and Applications

Use Inverse Matrices to Solve Linear Systems

The Inverse of a Square Matrix

Review Problems Matrices

Finding the Inverse of a Matrix

Matrix Operations SpringSemester 2017.

الوحدة السابعة : المصفوفات . تنظيم البيانات فى مصفوفات . الوحدة السابعة : المصفوفات . تنظيم البيانات فى مصفوفات . 1 جمع المصفوفات وطرحها.

Section 6.4 Multiplicative Inverses of Matices and Matrix Equations

Use Inverse Matrices to Solve Linear Systems

Solving Linear Systems Using Inverse Matrices

Using matrices to solve Systems of Equations

Use Inverse Matrices to Solve 2 Variable Linear Systems

Section 9.4 Multiplicative Inverses of Matices and Matrix Equations

Lesson 4: Inverses of Matrices (part 1)

Matrix Operations SpringSemester 2017.

The Identity and Inverse Matrices

Using matrices to solve Systems of Equations

Jeffrey Bivin Lake Zurich High School

Presentation transcript:

Inverse & Identity MATRICES Last Updated: October 12, 2005

Inverses and Identities 5x = 3

Now with Matrices ? This is the Identity Matrix for 2 x 2 Matrices

Let’s look at another example

What do we multiply a matrix by to get the Identity? New Question ? What do we multiply a matrix by to get the Identity?

The Inverse of a 2x2 Matrix

The Inverse of a 2x2 Matrix

The Inverse of a 2x2 Matrix

A B This is our Formula!

A B This is our Formula!

A B C This is our Formula!

Are the two Matrices Inverses? The product of inverse matrices is the identity matrix. Identity, therefore, INVERSE Matrices

Are the two Matrices Inverses? The product of inverse matrices is the identity matrix. Not the Identity, therefore, NOT INVERSE Matrices

Does the Matrix have an Inverse? Let’s review the definition of the Inverse of a 2x2 Matrix

The Inverse of a 2x2 Matrix

Does the Matrix have an Inverse? Find the determinant! Therefore, NO inverse!

Does the Matrix have an Inverse? Find the determinant! Therefore, an inverse exists!

Does the Matrix have an Inverse? + + 1 2 3 4 5 6 7 8 9 Find the determinant! - - - 1•5•9 + 2•6•7 + 3•4•8 - 7•5•3 - 8•6•1 - 9•4•2 45 + 84 + 96 - 105 - 48 - 72 Therefore, NO inverse!

Does the Matrix have an Inverse? + + 1 3 2 4 Find the determinant! - - - 1•4•2 + 3•1•3 + 2•2•4 - 3•4•2 - 4•1•1 - 2•2•3 8 + 9 + 16 - 24 - 4 - 12 -7 Therefore, an inverse exists!

Solve the system using inverse matrices B 3x + 2y = 7 4x - 5y = 11 This is our Formula!

Solve the system using inverse matrices B 2x - 4y = 9 3x - 2y = 1 This is our Formula!