Copyright © 2011 Pearson, Inc. 7.2 Matrix Algebra

Copyright © 2011 Pearson, Inc. Slide 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.

Copyright © 2011 Pearson, Inc. Slide Matrix

Copyright © 2011 Pearson, Inc. Slide Matrix Vocabulary Each element, or entry, a ij, of the matrix uses double subscript notation. The row subscript is the first subscript i, and the column subscript is j. The element a ij is the ith row and the jth column. In general, the order of an m  n matrix is m  n.

Copyright © 2011 Pearson, Inc. Slide Example Determining the Order of a Matrix

Copyright © 2011 Pearson, Inc. Slide Example Determining the Order of a Matrix

Copyright © 2011 Pearson, Inc. Slide Matrix Addition and Matrix Subtraction

Copyright © 2011 Pearson, Inc. Slide Example Matrix Addition

Copyright © 2011 Pearson, Inc. Slide Example Matrix Addition

Copyright © 2011 Pearson, Inc. Slide Example Using Scalar Multiplication

Copyright © 2011 Pearson, Inc. Slide Example Using Scalar Multiplication

Copyright © 2011 Pearson, Inc. Slide The Zero Matrix

Copyright © 2011 Pearson, Inc. Slide Additive Inverse

Copyright © 2011 Pearson, Inc. Slide Matrix Multiplication

Copyright © 2011 Pearson, Inc. Slide Example Matrix Multiplication

Copyright © 2011 Pearson, Inc. Slide Example Matrix Multiplication

Copyright © 2011 Pearson, Inc. Slide Example Matrix Multiplication

Copyright © 2011 Pearson, Inc. Slide Example Matrix Multiplication

Copyright © 2011 Pearson, Inc. Slide Identity Matrix

Copyright © 2011 Pearson, Inc. Slide Inverse of a Square Matrix

Copyright © 2011 Pearson, Inc. Slide Inverse of a 2 × 2 Matrix

Copyright © 2011 Pearson, Inc. Slide Determinant of a Square Matrix

Copyright © 2011 Pearson, Inc. Slide Inverses of n  n Matrices An n  n matrix A has an inverse if and only if det A ≠ 0.

Copyright © 2011 Pearson, Inc. Slide Example Finding Inverse Matrices

Copyright © 2011 Pearson, Inc. Slide Example Finding Inverse Matrices

Copyright © 2011 Pearson, Inc. Slide Example Finding Inverse Matrices

Copyright © 2011 Pearson, Inc. Slide Properties of Matrices Let A, B, and C be matrices whose orders are such that the following sums, differences, and products are defined. 1. Community 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·I n = I n ·A = A

Copyright © 2011 Pearson, Inc. Slide 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 -1 A = I n |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

Copyright © 2011 Pearson, Inc. Slide Quick Review

Copyright © 2011 Pearson, Inc. Slide Quick Review Solutions