2 5 Matrix Multiplication Size of the product

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2. 5 Matrix Multiplication Size of the product: If A is a matrix of

2. 5 Matrix Multiplication Size of the product: If A is a matrix of size m x n and B is a matrix of size n x p (note: the column size of A must equal the row size of B), then the product AB will be a matrix of size m x p. Example. Matrix A size 3 x 2 and matrix B size 2 x 5 The product AB will be a matrix of size 3 x 5 Example. Matrix A size 3 x 4 and matrix B size 3 x 4 The product AB can’t be computed Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

For matrices A, B, and C, let AB = C. Then And so on…

For matrices A, B, and C, let AB = C. Then And so on… Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Example. Given matrices B and C find BC and CB. Note: in general, for

Example. Given matrices B and C find BC and CB. Note: in general, for any two square matrices B and C Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Laws for Matrix Multiplication If the products and sums are defined for matrices A,

Laws for Matrix Multiplication If the products and sums are defined for matrices A, B, and C we have the Associative law (AB)C = A(BC) and the Distributive law: A(B + C) = AB + AC. The identity matrix of size n is given by Diagonal of 1’s In A = A and BIn = B where defined. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Matrix Equation Representation of a System of Linear Equations AX = B Copyright ©

Matrix Equation Representation of a System of Linear Equations AX = B Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Three network consultants, Alan, Maria, and Steven, each received a year-end bonus of $10,

Three network consultants, Alan, Maria, and Steven, each received a year-end bonus of $10, 000, which they decided to invest in a (401)K retirement plan sponsored by their employer. Under this plan, each employee is allowed to place their investments in three funds-an equity index fund (I), a growth fund (II), and a global equity fund (III). The allocations of the investments (in dollars) of the three employees at the beginning of the year are summarized in the matrix I II III Alan Maria Steven The return of the three funds after 1 yr is 18% for fund I, 24% for fund II, and 12% for fund III. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

a. Write a column vector B representing the returns of the three funds after

a. Write a column vector B representing the returns of the three funds after 1 year. b. Which employee realize the best returns on his or her investment for the year in questions? The worst return? Solution: (a) I II III Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

(b) return Alan Maria Steven Hence Maria realized the best returns on her investment

(b) return Alan Maria Steven Hence Maria realized the best returns on her investment ($1920); and Steven realized the worst returns on his investment ($1680). Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.