1. Matrix Lesson 2:
Cramer’s rule
Dave Coulson, 2013
2. Three men take turns standing in pairs on a weighing machine.
Alan and Barry together weigh 140kg.
Alan and Charlie together weigh 110kg.
Barry and Charlie together weigh 90kg.
How heavy are the three men individually?
3. In maths, the problem looks like this: A + B = 140
A + C = 110
B + C = 90
This is a 3x3 system of linear equations.
“3x3” because there are three equations and three unknowns.
“Linear” because the unknowns are simply being added together.
There are no squares or square roots or other peculiar functions.
4. I‟m going to solve this system
using a procedure called
Cramer’s Rule.
A + B = 140
A + C = 110
B + C = 90
It‟s quite longwinded, but I like it because it produces the answer
in a seemingly magical way.
For most of the journey you won‟t know what you‟re doing or
why you‟re doing it. but then the answer suddenly pops out in
the most unlikely way.
5. First, rewrite the system in matrix notation:
=
1 1 0
1 0 1
0 1 1
A
B
C
140
110
90
A + B = 140
A + C = 110
B + C = 90
6. Then get the determinant of the matrix.
=
1 1 0
1 0 1
0 1 1
A
B
C
140
110
90
Detm = -2
1 1 0 1 1
1 0 1 1 0
0 1 1 0 1
(0 + 0 + 0) - (0 + 1 + 1)
7. Do it again, but this time replace the „A‟ row of the matrix
with the numbers from the right-hand-side of the equation.
=
1 1 0
1 0 1
0 1 1
A
B
C
140
110
90
Detm = -2
140 1 0 140 1
110 0 1 110 0
90 1 1 90 1
DetA = -160
8. Detm = -2
A = 80kg
=
1 1 0
1 0 1
0 1 1
A
B
C
140
110
90
DetA = -160
The value for A is the fraction made from
these two numbers (with DetA on top).
9. Detm = -2
B and C can be worked out in the same sort of way.
Replace the „B‟ line of the matrix with the numbers
on the right hand side of the equation.
=
1 1 0
1 0 1
0 1 1
A
B
C
140
110
90
(next screen)
13. =
1 1 0
1 0 1
0 1 1
A
B
C
140
110
90
Detm = -2
DetC = -60
C = 30kg
14. Always check the answers against the initial clues:
Alan and Barry together weigh 140kg.
Alan and Charlie together weigh 110kg.
Barry and Charlie together weigh 90kg.
80 + 60
80 + 30
60 + 30
(Charlie is obviously a child)
15. [END]
Dave Coulson, 2013
According to Wikipedia, the technique
was established by Gabriel Cramer (a
Swiss mathematician) in 1750, “although
Colin MacLaurin [Scottish] also
published special cases of the rule in
1748 (and possibly knew of it as early as
1729).”
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