Precalculus Lesson 7.2 Matrix Algebra 4/6/2017 8:43 PM

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Precalculus Lesson 7.2 Matrix Algebra 4/6/2017 8:43 PM
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Quick Review

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

Example Using Scalar Multiplication

The Zero Matrix Example:

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