- #1

Hypatio

- 151

- 1

I want to solve the following system of equations

##M_{1} = f_1+f_2+m_1+m_2\ \ ;\ \ M_{7} = f_1+f_2+s_1+s_2\ \ ;\ \ M_{13} = m_1+m_2+s_1+s_2##

##M_{2} = f_1+f_3+m_1+m_3\ \ ;\ \ M_{8} = f_1+f_3+s_1+s_3\ \ ;\ \ M_{14} = m_1+m_3+s_1+s_3##

##M_{3} = f_1+f_4+m_1+m_4\ \ ;\ \ M_{9} = f_1+f_4+s_1+s_4\ \ ;\ \ M_{15} = m_1+m_4+s_1+s_4##

##M_{4} = f_2+f_3+m_2+m_3\ \ ;\ \ M_{10} = f_2+f_3+s_2+s_3\ \ ;\ \ M_{16} = m_2+m_3+s_2+s_3##

##M_{5} = f_2+f_4+m_2+m_4\ \ ;\ \ M_{11} = f_2+f_4+s_2+s_4\ \ ;\ \ M_{17} = m_2+m_4+s_2+s_4##

##M_{6} = f_3+f_4+m_3+m_4\ \ ;\ \ M_{12} = f_3+f_4+s_3+s_4\ \ ;\ \ M_{18} = m_3+m_4+s_3+s_4##

where the 18 M coefficients are known and the 12 f, m, and s coefficients are unknown.

The strategy I want to use is to build a coefficient matrix and solve by inversion, but how can I do this for this system with minimal algebraic manipulation? For example, it is easy to see that at least 6 expressions for each unknown coefficient can be written. Perhaps this could result in a (6*12) by (6*12) coefficient matrix, but I'm unsure what it would look like, or if I am thinking about this correctly.

So, how do I arrange the coefficients so that

##f_1=M_1-f_2-m_1-m_2##

and

##f_1=M_2-f_3-m_1-m_3##

etc. are both satisfied. I thought you could make them both separate variables, like ##f_{1a}## and ##f_{1b}## and then have another equation express that ##f_{1a}=f_{1b}##, but I can't recall how to do this.

##M_{1} = f_1+f_2+m_1+m_2\ \ ;\ \ M_{7} = f_1+f_2+s_1+s_2\ \ ;\ \ M_{13} = m_1+m_2+s_1+s_2##

##M_{2} = f_1+f_3+m_1+m_3\ \ ;\ \ M_{8} = f_1+f_3+s_1+s_3\ \ ;\ \ M_{14} = m_1+m_3+s_1+s_3##

##M_{3} = f_1+f_4+m_1+m_4\ \ ;\ \ M_{9} = f_1+f_4+s_1+s_4\ \ ;\ \ M_{15} = m_1+m_4+s_1+s_4##

##M_{4} = f_2+f_3+m_2+m_3\ \ ;\ \ M_{10} = f_2+f_3+s_2+s_3\ \ ;\ \ M_{16} = m_2+m_3+s_2+s_3##

##M_{5} = f_2+f_4+m_2+m_4\ \ ;\ \ M_{11} = f_2+f_4+s_2+s_4\ \ ;\ \ M_{17} = m_2+m_4+s_2+s_4##

##M_{6} = f_3+f_4+m_3+m_4\ \ ;\ \ M_{12} = f_3+f_4+s_3+s_4\ \ ;\ \ M_{18} = m_3+m_4+s_3+s_4##

where the 18 M coefficients are known and the 12 f, m, and s coefficients are unknown.

The strategy I want to use is to build a coefficient matrix and solve by inversion, but how can I do this for this system with minimal algebraic manipulation? For example, it is easy to see that at least 6 expressions for each unknown coefficient can be written. Perhaps this could result in a (6*12) by (6*12) coefficient matrix, but I'm unsure what it would look like, or if I am thinking about this correctly.

So, how do I arrange the coefficients so that

##f_1=M_1-f_2-m_1-m_2##

and

##f_1=M_2-f_3-m_1-m_3##

etc. are both satisfied. I thought you could make them both separate variables, like ##f_{1a}## and ##f_{1b}## and then have another equation express that ##f_{1a}=f_{1b}##, but I can't recall how to do this.

Last edited: