# MATLAB: How to solve a system of 9 linear equations efficiently? (long computation time)

efficiencylong computation timeMATLABmatrix multiplicationsolvesymbolicsymbolic equationsystem of linear equations

I got a system of 9 linear equations which I received from a matrix multiplication. I want to solve this system efficiently for all 9 variables, altough I got some more symbolic variables in each equation. I was thinking of using a loop to solve this problem, but since I'm a newbie in Matlab, I'm not quite sure which method works best or how to solve this problem in an elegant way.
``%% definitionssyms WC WP; % parameterssyms D1; %variablesyms G1 G2; % parameters%h = 1.0545717*10^(-34);h=1;% variablessyms a1 a2 a3;syms b1 b2 b3;syms c1 c2 c3;% matrices usedH=-h*0.5*[0 0 WP;    0 2*D1 WC;    WP WC 2*D1];R=[a1 a2 a3;    b1 b2 b3;    c1 c2 c3];B=[G1*c3 0 -0.5*a3*(G1 + G2);    0 G2*c3 -0.5*b3*(G1 + G2);    -0.5*c1*(G1 + G2) -0.5*c2*(G1 + G2) -c3*(G1 + G2)];%% calculationdtR = -1i/h*(H*R - R*H)+B == 0;% extracting "equations" from the matrix result dtR and asigning a variable/equationeqn1 = dtR(1,1);eqn2 = dtR(1,2);eqn3 = dtR(1,3);    % <---- this should be doable in a loop?eqn4 = dtR(2,1);eqn5 = dtR(2,2);eqn6 = dtR(2,3);eqn7 = dtR(3,1);eqn8 = dtR(3,2);eqn9 = dtR(3,3);eqns = [eqn1,eqn2,eqn3,eqn4,eqn5,eqn6,eqn7,eqn8,eqn9];S = solve(eqns, [a1 a2 a3 b1 b2 b3 c1 c2 c3], 'ReturnConditions', true); % <--- this takes very long to computeS.c1,S.conditions,S.parameters,``
So the equations I get are a bit mixed up, but that shouldn't matter for solving this system of linear equations. However Matlab takes a really long time to compute this, so I hope someone has a tip or advice how I can solve this problem in a more efficient way.

``M = sym('M', [rows, columns])S = M\B;S = subs(S, M, 1i/h*(H*R - R*H))``