# Analytic inversion of linear system [closed]

My question is at the frontier of maths and mathematica, but the main point is to ask if it is feasible in mathematica and which functions can help me to do it, that is why I posted it here.

I have an invertible linear system of equations.

I don't exactly know the number of equations I have (in fact I haven't finished the preliminary work to get to it), but to give an Idea it will approximatively be 20 equations.

I can write my problem like this :

$$A.X = B$$

To solve it I could invert the matrix $A$ to get :

$$X=A^{-1} B$$

But I am pretty sure that it would be too hard for mathematica (I recall that it is not a numerical problem, but an analytic one, my matrix is full of unknown variables).

In fact I am only interested in the first line of my system : $X(1)$ (maybe it can help to solve the problem).

Thus, I am wondering if there is a better mathematical method to help me to do it. I have heard about LU decomposition but maybe there is an even better method as I am just interested in $X(1)$ ? (I don't know very well LU decomposition, I just heard it is "faster" than a matrix inversion but I don't know much more)

And if there is such method, my main question is : which mathematica functions can help me to do it ?

Finally :

• What is the most efficient method to analytically solve my problem : I only need to get $X(1)$ in my system ?
• How is it implemented in Mathematica ?
• Do you think that for a linear system of 20-30 equations it is feasible with mathematica ? Or it is too hard ? (I have a fast laptop but I am not working on supercomputer for example).

## closed as off-topic by m_goldberg, MarcoB, LCarvalho, LLlAMnYP, bbgodfreyJun 14 '17 at 13:08

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question cannot be answered without additional information. Questions on problems in code must describe the specific problem and include valid code to reproduce it. Any data used for programming examples should be embedded in the question or code to generate the (fake) data must be included." – MarcoB, LLlAMnYP, bbgodfrey
If this question can be reworded to fit the rules in the help center, please edit the question.

• Whether it is feasible or not depends on the sparsity of your matrix. You might want to look at LinearSolve, which solves linear equations without fully inverting the matrix. – yohbs Jun 8 '17 at 1:23
• When you say $X(1)$, do you mean the first row of X or the first column? – bill s Jun 8 '17 at 3:00
• The first row. X is a column vector – StarBucK Jun 8 '17 at 10:01