Solving Matrix Equations with Multiple Variable Vectors

I have an equation: S = QE

where Q is a 3x3 matrix with known constants,

S and E are both 3 element vectors. S has a single variable and E has 2 variables.

I want to solve for the 3 variables. This is what I tried:

E1 = 140e9;    % [Pa]
E2 = 10e9;     % [Pa]
G12 = 7e9;     % [Pa]
V12 = 0.3;
V21 = V12;

syms tau Exx Eyy

S = [tau + 5e6;...  % S11
    -tau + 5e6;...  % S22
       -5e6];       % S12

Q = [E1/(1-V12*V21) V12*E2/(1-V12*V21) 0;... % All known values
  V12*E2/(1-V12*V21)  E2/(1-V12*V21)   0;...
         0                 0           0];

E = [Exx/2 + Eyy/2;...
     Exx/2 + Eyy/2;...
       Exx - Eyy];

solve(S == Q*E);

But it comes up empty. I've done this same process with a set of equations, but never in matrix form. I could, of course, expand this out by hand and input the seperate equations, but I'd rather not have to do that because I'd like to be able to turn this into a function. I'm pretty sure linsolve() won't work since the variable are not spread between 2 vectors. Not sure what else to try.

Any suggestions are appreciated, thanks.

[Stephan]

Are you sure that your matrix is written correctly?

Q =

   1.0e+11 *

    1.5385    0.0330         0
    0.0330    0.1099         0
         0         0         0

[nick12391]

I'm not sure of anything, Stephen, But I will triple-check everything.

[nick12391]

I found my mistake. I should have had Q33 = G12. I have been struggling with this all day, and all along it was just a typo -_-

[Stephan]

https://de.wikipedia.org/wiki/Unidirektionale_Schicht

E1 = 140e9;    % [Pa]
E2 = 10e9;     % [Pa]
G12 = 7e9;     % [Pa]
V12 = 0.3;
V21 = V12;

syms tau Exx Eyy

S = [tau + 5e6;...  % S11
    -tau + 5e6;...  % S22
       -5e6];       % S12

Q = [E1/(1-V12*V21) V12*E2/(1-V12*V21) 0;... % All known values
  V12*E2/(1-V12*V21)  E2/(1-V12*V21)   0;...
         0                 0           G12];

E = [Exx/2 + Eyy/2;...
     Exx/2 + Eyy/2;...
       Exx - Eyy];

S = (Q\E)

[nick12391]

Ja das ist ein sieg. Thanks again!

[Stephan]

I think you have one more bug in your matrix. Should not Q(2,1) be:

V12*E1/(1-V12*V21)

instead of:

V12*E2/(1-V12*V21)

due to

https://wikimedia.org/api/rest_v1/media/math/render/svg/acf95d9b3a35507b5466445d80c7e5d6535758c7

[nick12391]

I think that would technically make it:

V21*E1/(1-V12*V21)

but since V12/E1 = V21/E2 it amounts to the same thing.

[Stephan]

But the thing is that you get a different result for tau, by doing this the other way. Or am i wrong here?

[nick12391]

This is how it is written in my textbook:

[Stephan]

Ok, then it should be done.