Lately I've been studying how to solve a system of linear equations; in particular, three linear equations. Not satisfied with the "Elimination of Variables" method for solving this sort of problem, I researched other ways of solving for x, y, and z values. I found that the "Rule of Sarrus" and "Cramer's Rule" are ideal for my purposes, and put these concepts into a Python script that would illustrate the rules. One can download the source code for this at: https://github.com/mikequentel/sarrus_cramer
Quick background on Rule of Sarrus and Cramer's Rule...
- Rule of Sarrus computes the determinant of a 3x3 matrix.
- Cramer's Rule solves a system of linear equations (where number of equations is equal to number of unknowns) using determinants (which can be obtained from the above Rule of Sarrus). Purple Math has an excellent explanation and use case for Cramer's Rule at: http://www.purplemath.com/modules/cramers.htm
- For a system of three linear equations, one can use the above rules to provide a solution, summarised as the following:
x = det(x)/det(coefficients)
y = det(y)/det(coefficients)
z = det(z)/det(coefficients)
I must mention that the code below is very simple, and lacks error-trapping (for example, nothing traps for division by zero). Reason for this rough implementation is to keep the code readable as an example of using the rules of Sarrus and Cramer, rather than an actual implementation that would be used in a real system.
# System of three linear equations
# ax + by + cz = j
# dx + ey + fz = k
# gx + hy + iz = l
# System of three linear equations in matrix notation
# - - - - - -
# | a b c | | x | | j |
# | | | | | |
# | d e f | | y | = | k |
# | | | | | |
# | g h i | | z | | l |
# - - - - - -
# Matrix of Coefficients
# a b c
# d e f
# g h i
# Matrix of Variables
# x
# y
# z
# Matrix of Resulting Values
# j
# k
# l
# Rule of Sarrus
# a b c|a b
# d e f|d e
# g h i|g h
# Rule of Sarrus Index Values
# 0 1 2|0 1
# 3 4 5|3 4
# 6 7 8|6 7
# Determinant
# det(M) = aei + bfg + cdh - gec - hfa - idb
# Cramer's Rule
# | j b c | | a j c | | a b j |
# | k e f | | d k f | | d e k |
# | l h i | | g l i | | g h l |
# ---------, ---------, ---------
# | a b c | | a b c | | a b c |
# | d e f | | d e f | | d e f |
# | g h i | | g h i | | g h i |
import sys
def main():
inputs_dict = {'a':int(raw_input("a:")), 'b':int(raw_input("b:")), 'c':int(raw_input("c:")), 'j':int(raw_input("j:")),
'd':int(raw_input("d:")), 'e':int(raw_input("e:")), 'f':int(raw_input("f:")), 'k':int(raw_input("k:")),
'g':int(raw_input("g:")), 'h':int(raw_input("h:")), 'i':int(raw_input("i:")), 'l':int(raw_input("l:"))}
coeffs_matrix = {'a':inputs_dict['a'], 'b':inputs_dict['b'], 'c':inputs_dict['c'],
'd':inputs_dict['d'], 'e':inputs_dict['e'], 'f':inputs_dict['f'],
'g':inputs_dict['g'], 'h':inputs_dict['h'], 'i':inputs_dict['i']}
x_numerator_matrix = {'j':inputs_dict['j'], 'b':inputs_dict['b'], 'c':inputs_dict['c'],
'k':inputs_dict['k'], 'e':inputs_dict['e'], 'f':inputs_dict['f'],
'l':inputs_dict['l'], 'h':inputs_dict['h'], 'i':inputs_dict['i']}
y_numerator_matrix = {'a':inputs_dict['a'], 'j':inputs_dict['j'], 'c':inputs_dict['c'],
'd':inputs_dict['d'], 'k':inputs_dict['k'], 'f':inputs_dict['f'],
'g':inputs_dict['g'], 'l':inputs_dict['l'], 'i':inputs_dict['i']}
z_numerator_matrix = {'a':inputs_dict['a'], 'b':inputs_dict['b'], 'j':inputs_dict['j'],
'd':inputs_dict['d'], 'e':inputs_dict['e'], 'k':inputs_dict['k'],
'g':inputs_dict['g'], 'h':inputs_dict['h'], 'l':inputs_dict['l']}
# Rule of Sarrus for det_coeffs_matrix
# a b c|a b
# d e f|d e
# g h i|g h
#
det_coeffs_matrix = (coeffs_matrix['a'] * coeffs_matrix['e'] * coeffs_matrix['i'] +
coeffs_matrix['b'] * coeffs_matrix['f'] * coeffs_matrix['g'] +
coeffs_matrix['c'] * coeffs_matrix['d'] * coeffs_matrix['h'] -
coeffs_matrix['g'] * coeffs_matrix['e'] * coeffs_matrix['c'] -
coeffs_matrix['h'] * coeffs_matrix['f'] * coeffs_matrix['a'] -
coeffs_matrix['i'] * coeffs_matrix['d'] * coeffs_matrix['b'])
# Rule of Sarrus for det_x_numerator_matrix
# j b c|j b
# k e f|k e
# l h i|l h
#
det_x_numerator_matrix = (x_numerator_matrix['j'] * x_numerator_matrix['e'] * x_numerator_matrix['i'] +
x_numerator_matrix['b'] * x_numerator_matrix['f'] * x_numerator_matrix['l'] +
x_numerator_matrix['c'] * x_numerator_matrix['k'] * x_numerator_matrix['h'] -
x_numerator_matrix['l'] * x_numerator_matrix['e'] * x_numerator_matrix['c'] -
x_numerator_matrix['h'] * x_numerator_matrix['f'] * x_numerator_matrix['j'] -
x_numerator_matrix['i'] * x_numerator_matrix['k'] * x_numerator_matrix['b'] )
# Rule of Sarrus for det_y_numerator_matrix
# a j c|a j
# d k f|d k
# g l i|g l
#
det_y_numerator_matrix = (y_numerator_matrix['a'] * y_numerator_matrix['k'] * y_numerator_matrix['i'] +
y_numerator_matrix['j'] * y_numerator_matrix['f'] * y_numerator_matrix['g'] +
y_numerator_matrix['c'] * y_numerator_matrix['d'] * y_numerator_matrix['l'] -
y_numerator_matrix['g'] * y_numerator_matrix['k'] * y_numerator_matrix['c'] -
y_numerator_matrix['l'] * y_numerator_matrix['f'] * y_numerator_matrix['a'] -
y_numerator_matrix['i'] * y_numerator_matrix['d'] * y_numerator_matrix['j'])
# Rule of Sarrus for det_z_numerator_matrix
# a b j|a b
# d e k|d e
# g h l|g h
#
det_z_numerator_matrix = (z_numerator_matrix['a'] * z_numerator_matrix['e'] * z_numerator_matrix['l'] +
z_numerator_matrix['b'] * z_numerator_matrix['k'] * z_numerator_matrix['g'] +
z_numerator_matrix['j'] * z_numerator_matrix['d'] * z_numerator_matrix['h'] -
z_numerator_matrix['g'] * z_numerator_matrix['e'] * z_numerator_matrix['j'] -
z_numerator_matrix['h'] * z_numerator_matrix['k'] * z_numerator_matrix['a'] -
z_numerator_matrix['l'] * z_numerator_matrix['d'] * z_numerator_matrix['b'])
x = det_x_numerator_matrix/det_coeffs_matrix
y = det_y_numerator_matrix/det_coeffs_matrix
z = det_z_numerator_matrix/det_coeffs_matrix
print
print "results: "
print "x = " + str(x)
print "y = " + str(y)
print "z = " + str(z)
# Specifies name of main function.
if __name__ == "__main__":
sys.exit(main())
