Precalculus Lesson 7.2 Matrix Algebra 4/6/2017 8:43 PM © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

Quick Review

What you’ll learn about Matrices Matrix Addition and Subtraction Matrix Multiplication Identity and Inverse Matrices Determinant of a Square Matrix Applications … and why Matrix algebra provides a powerful technique to manipulate large data sets and solve the related problems that are modeled by the matrices.

Matrix

Matrix Vocabulary Each element, or entry, aij, of the matrix uses double subscript notation. The row subscript is the first subscript i, and the column subscript is j. The element aij is the ith row and the jth column. In general, the order of an m × n matrix is m×n.

Example Determining the Order of a Matrix

Matrix Addition and Matrix Subtraction

Example Matrix Addition

Example Using Scalar Multiplication

The Zero Matrix Example:

Additive Inverse

Example Using Additive Inverse

Matrix Multiplication

Example Matrix Multiplication

Identity Matrix

Inverse of a Square Matrix

Example Inverse of a Square Matrices Yes

Inverse of a 2 × 2 Matrix

Determinant of a Square Matrix Refer to text pg 583

Inverses of n × n Matrices An n × n matrix A has an inverse if and only if det A ≠ 0.

Example Finding Inverse Matrices

Properties of Matrices Let A, B, and C be matrices whose orders are such that the following sums, differences, and products are defined. 1. Commutative property Addition: A + B = B + A Multiplication: Does not hold in general 2. Associative property Addition: (A + B) + C = A + (B + C) Multiplication: (AB)C = A(BC) 3. Identity property Addition: A + 0 = A Multiplication: A·In = In·A = A

Properties of Matrices Let A, B, and C be matrices whose orders are such that the following sums, differences, and products are defined. 4. Inverse property Addition: A + (-A) = 0 Multiplication: AA-1 = A-1A = In |A|≠0 5. Distributive property Multiplication over addition: A(B + C) = AB + AC (A + B)C = AC + BC Multiplication over subtraction: A(B - C) = AB - AC (A - B)C = AC - BC

Homework: Text pg588/589 Exercises #2, 4, 14, 20, 24, and 34