Check even answers p.76: Hint: One of the problems on p.76 has

Slides:

Advertisements

Similar presentations

Determinant The numerical value of a square array of numbers that can be used to solve systems of equations with matrices. Second-Order Determinant (of.

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

4.5 Inverses of Matrices.

Inverses of n x n Matrices. The Inverse Matrix If A is an n x n matrix, the inverse of A (call it A -1 ) is the matrix such that A * A -1 is equal to.

Section 9.6 Determinants and Inverses Objectives To understand how to find a determinant of a 2x2 matrix. To understand the identity matrix. Do define.

4-5 Matrix Inverses and Solving Systems Warm Up Lesson Presentation

Other types of Equations Polynomial Equations. Factoring Solve.

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.

Section 10.3 – The Inverse of a Matrix No Calculator.

4.7 Inverse Matrices and Systems. 1) Inverse Matrices and Systems of Equations You have solved systems of equations using graphing, substitution, elimination…oh.

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.

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

VAVLPVCTYMAUS PSABLADDERZSB EBSANTESHTICL RLDUDSKTTVSRA EDEARCENEAUOD CRFNORSASINTD TPEUUOCPTDATP UNRTMTRBEEXME MIEUSUULSNSNN USNMEMNISAIIT AESXSVPENNISI.

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.

Spring 2013 Solving a System of Linear Equations Matrix inverse Simultaneous equations Cramer’s rule Second-order Conditions Lecture 7.

QPLNHTURBIOTS CADAIASOINCOS OSTPOSTLGVAGT AJRLFKLEROUEA CLARITYSOLSTB HTEAMVSRUVAHI INTERACTPELEL NAPKSOCIALIRI GSOCIOGRAMTST CONFORMITYYTY 14 WORDS ANSWERS.

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.

Using Matrices to Solve Systems of Equations Matrix Equations l We have solved systems using graphing, but now we learn how to do it using matrices.

Chapter 4 Matrices By: Matt Raimondi.

Inverse Matrices and Systems

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

14.3 Matrix Equations and Matrix Solutions to 2x2 Systems OBJ: Use the Inverse of a 2 x 2 Matrix to solve a system of equations.

Holt Algebra Matrix Inverses and Solving Systems A matrix can have an inverse only if it is a square matrix. But not all square matrices have inverses.

Objectives Determine whether a matrix has an inverse.

Identity and Inverse Matrices. Key Topics Identity matrix: a square matrix, multiplied with another matrix doesn’t change the other matrix (just like.

Inverse & Identity Matrices

Find the determinant of this matrix Jeff buys 7 apples and 4 pears for $7.25. At the same prices, Hayley buy 5 apples and 9 pears for $10.40.

2.5 Determinants and Multiplicative Inverses of Matrices Objectives: Evaluate determinants. Find inverses of matrices. Solve systems of equations by using.

1. Inverse of A 2. Inverse of a 2x2 Matrix 3. Matrix With No Inverse 4. Solving a Matrix Equation 1.

EXAMPLE 3 Find the inverse of a 3 × 3 matrix Use a graphing calculator to find the inverse of A. Then use the calculator to verify your result. 2 1 – 2.

Warm-Up 3) Find the determinant by hand. 4) Find the determinant using your calculator. 1) Multiply. Show work. 2) Multiply. Show work.

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

Using square roots to solve quadratic equations. 2x² = 8 22 x² = 4 The opposite of squaring a number is taking its square root √ 4= ± 2.

Section 10.3 and Section 9.3 Systems of Equations and Inverses of Matrices.

4.8 Using matrices to solve systems! 2 variable systems – by hand 3 or more variables – using calculator!

Find the determinate of both of the following matrices.

Some examples for identity matrixes are I 1 =, I 2 =, I 3 = … Let’s analyze the multiplication of any matrix with identity matrix. = We can conclude that.

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

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

2.5 – Determinants and Multiplicative Inverses of Matrices.

Worksheet Answers Matrix worksheet And Matrices Review.

Warm Up Multiple the matrices. 1. Find the determinant –1 0.

Section 6-2: Matrix Multiplication, Inverses and Determinants There are three basic matrix operations. 1.Matrix Addition 2.Scalar Multiplication 3.Matrix.

Chapter 4 Matrices.

3.1 Introduction to Determinants

Review Problems Matrices

Unit 1: Matrices Day 1 Aug. 7th, 2012.

Finding the Inverse of a Matrix

Solving Matrix equations

Complete the missing numbers using the inverse.

4-5 Matrix Inverses and Solving Systems Warm Up Lesson Presentation

Multiplying Matrices.

More about Matrices Chapter 4, Sections 5, 7.

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

Fundamentals of Engineering Analysis

Section 3.3 – The Inverse of a Matrix

ARRAY DIVISION Identity matrix Islamic University of Gaza

4.4 Objectives Day 1: Find the determinants of 2  2 and 3  3 matrices. Day 2: Use Cramer’s rule to solve systems of linear equations. Vocabulary Determinant:

Bellwork 1) Multiply. 3) Find the determinant. 2) Multiply.

Inverse Matrices and Systems

Lesson 6: Inverse of Matrices (part 3)

Matrices and Determinants

The Identity and Inverse Matrices

L4-5/L4-6 Objective: Students will be able to evaluate determinants of matrices.

Presentation transcript:

Check even answers p.76: Hint: One of the problems on p.76 has (2, -7, 12) (-5, 2, 4) (1/2, -3/4, 1/5) (-16, 24, 12) 28. 2π (show work!) Hint: One of the problems on p.76 has no solution. Why is there no solution??

The following problems from pages 82-84 were not possible: #9, 12, 29, 39

Notes 2-5 Determinants & Inverses Each square matrix has a determinant, which is a single numerical value. If the determinant is 0, then the inverse of the matrix does not exist (undefined) 2 x 2 determinant (2nd order)

3 x 3 determinant (3rd order) “expansion by minors” 4

I = Identity Matrices: 2nd order  I = 3rd order 4th order I = I =

The product of a matrix (A) and its inverse is the identity matrix: Determinant 6