1. 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
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2. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
For matrices A, B, and C, let AB = C.
Then
And so on…
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3. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Example. Given matrices B and C find BC and CB.
Note: in general, for any two square matrices B and C
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4. 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.
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Matrix Equation Representation of a System of Linear Equations
AX = B
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6. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Three network consultants, Alan, Maria, and Steven, each received a year-end bonus of $10,000, 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.
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Write a column vector B representing the returns of the three funds after 1 year.
Which employee realize the best returns on his or her investment for the year in questions? The worst return?
Solution: (a) I II
III
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8. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
return
Alan
Maria
Steven
Hence Maria realized the best returns on her investment ($1920); and
Steven realized the worst returns on his investment ($1680).
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