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Omnoculus

Posts: 2
From: New York State
Registered: 2/18/10

A textbook problem misprint?
Posted: Feb 18, 2010 5:42 PM

Plain Text

I have this textbook "Mathematics for 3D Game Programming & Computer Graphics" and Chapter on matrices starts from a linear algebra perspective. They have this example problem of a linear system that that go through and reduce to canonical form using an algorithm. I have been going over it again and again and I cannot come up with the same answer they get in the book for one operation in a particular element. The weird thing is the rest of the operations end up with the right solution for the system.

I'll repro the work here:

Given system: 3x + 2y - 3z = -13
4x - 3y + 6z = 7
x - z = -5

Augmented matrix:

| 3 2 -3 | -13 |
| 4 -3 6 | 7 |
| 1 0 -1 | -5 |

First swap rows one and two and multiply new row 1 by 1/4:

| 1 -3/4 3/2 | 7/4 |
| 3 2 -3 | -13 |
| 1 0 -1 | -5 |

Next make row 2 = -3 * (row 1) + row 2
and make row 3 = -1 * (row 1) + row 3 :

| 1 -3/4 3/2 | 7/4 |
| 0 17/4 -15/2 | -73/4 |
| 0 3/4 -5/2 | -27/4 |

Multiply row 2 by 4/17:

| 1 -3/4 3/2 | 7/4 |
| 0 1 -30/17 | -73/17 |
| 0 3/4 -5/2 | -27/4 |

Here's the step I have a problem with -
-2/3R2 + R1 -> R1 and
-3/4R2 + R3 -> R3

| 1 0 3/17 | -25/17 |
| 0 1 -30/17 | -73/17 |
| 0 0 -20/17 | -60/17 |

My problem is with the -25/17 in element 1,4
The operation there should be ( -2/3 * -73/17 ) + 7/4
unless I am wrong. But then how could the answer be negative? The answer I get is 941/204 but using this does not ultimately solve the system. I'll show work below but the last steps with their data:

Multiply Row 3 by -17/20:

| 1 0 3/17 | -25/17 |
| 0 1 -30/17 | -73/17 |
| 0 0 1 | 3 |

Finally -3/17R3 + R1 -> R1
and 30/17R3 + R2 -> R2 :

| 1 0 0 | -2 | x = -2
| 0 1 0 | 1 | y = 1
| 0 0 1 | 3 | z = 3

These values plugin to the equations correctly. So what am I missing? For -2/3R2 + R1 -> R1 get the following:

(-2/3 * -73/17) + 7/4
= 146/51 + 7/4
= 584/204 + 357/204
= 941/204

Any help is appreciated