Ralph Vince - Portfolio Mathematics

256 THE HANDBOOK OF PORTFOLIO MATHEMATICS

and the sixth row from the starting augmented matrix. We then use row
operations to perform elementary transformations until, again, the identity
matrix is obtained:

Starting Augmented Matrix

X 1 X 2 X 3 L 1 L 2 | Answer

0.095 0.13 0.21 0 0 | 0.18

1 1100| 1

0.1 −0.023 0.01 0.095 1 | 0

−0.023 0.25 0.079 0.13 1 | 0

0.01 0.079 0.4 0.21 1 | 0

Through the use of row operations...the identity matrix is obtained:

10000 | 0.1283688 =X 1

01000 | 0.1904699 =X 2

00100 | 0.6811613 =X 3

00010 |−2.38/.5=−4.76 =L 1

00001 | 0.210944/.5=.4219=L 2

When you must pull out a row and column like this, it is important that
you remember what rows correspond to what variables, especially when
you have more than one row and column to pull. Again, using an example to
illustrate, suppose we want to solve for E=.1965. The first identity matrix
we arrive at will show negative values for the weighting of Toxico, X 1 , and
the savings account, X 4. Therefore, we return to our starting augmented
matrix:

Starting Augmented Matrix

X 1 X 2 X 3 X 4 L 1 L 2 | Answer Pertains to

0.095 0.13 0.21 0.085 0 0 | 0.1965 Toxico
1 11100| 1 Incubeast
0.1 −0.023 0.01 0 0.095 1 | 0 LA Garb
−0.023 0.25 0.079 0 0.13 1 | 0 Savings
0.01 0.079 0.4 0 0.21 1 | 0L 1
0 0 0 0 0.085 1 | 0L 2