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**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.

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Quick Review

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**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.

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Matrix

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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.

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**Example Determining the Order of a Matrix**

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**Matrix Addition and Matrix Subtraction**

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**Example Matrix Addition**

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**Example Using Scalar Multiplication**

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The Zero Matrix Example:

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Additive Inverse

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**Example Using Additive Inverse**

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**Matrix Multiplication**

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**Example Matrix Multiplication**

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Identity Matrix

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**Inverse of a Square Matrix**

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**Example Inverse of a Square Matrices**

Yes

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Inverse of a 2 × 2 Matrix

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**Determinant of a Square Matrix**

Refer to text pg 583

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**Inverses of n × n Matrices**

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

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**Example Finding Inverse Matrices**

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**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

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**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

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

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