# Why these equations are not giving any solution? [closed]

I am trying to solve these three coupled linear equations but this gives me zero value. I don't know why it is happening. If anyone can resolve this is most welcome.

p11 = 1;
omega = .2;
C11 = 2.030;
beta = 1;
C55 = 0.600;
alpa = 1;
C13 = 0.750;
e31 = 0.200;
e15 = 3.700;
C33 = 2.450;
e33 = 1.300;
ep11 = 38.90;
ep33 = 25.70;
NSolve[{(p11*omega^2 - C11*beta^2 - C55*alpa^2)*A1 - (C13 + C55)*alpa*
beta*B1 - (e31 + e15)*alpa*beta*C1 ==
0, (p11*omega^2 - C55*beta^2 - C33*alpa^2)*B1 - (C13 + C55)*alpa*
beta*A1 - (e15*beta^2 + e33*alpa^2)*C1 ==
0, (ep11*beta^2 + ep33*alpa^2)*C1 - (e15*beta^2 + e33*alpa^2)*
B1 - (e15 + e31)*alpa*beta*A1 == 0}, {A1, B1, C1}]

• I’m voting to close this question because the issued raised is not really a problem; it arises from the OP not understanding of the result returned by Mathematica. – m_goldberg Feb 24 at 2:33

## 2 Answers

{0, 0, 0} is the solution

Clear["Global*"]

p11 = 1;
omega = .2;
C11 = 2.030;
beta = 1;
C55 = 0.600;
alpa = 1;
C13 = 0.750;
e31 = 0.200;
e15 = 3.700;
C33 = 2.450;
e33 = 1.300;
ep11 = 38.90;
ep33 = 25.70;

eqns = {(p11*omega^2 - C11*beta^2 - C55*alpa^2)*A1 - (C13 + C55)*alpa*beta*
B1 - (e31 + e15)*alpa*beta*C1 ==
0, (p11*omega^2 - C55*beta^2 - C33*alpa^2)*B1 - (C13 + C55)*alpa*beta*
A1 - (e15*beta^2 + e33*alpa^2)*C1 ==
0, (ep11*beta^2 + ep33*alpa^2)*C1 - (e15*beta^2 + e33*alpa^2)*
B1 - (e15 + e31)*alpa*beta*A1 == 0};

Solve[Rationalize@eqns, {A1, B1, C1}]

(* {{A1 -> 0, B1 -> 0, C1 -> 0}} *)


Graphically,

ContourPlot3D[Evaluate@eqns, {A1, -1, 1}, {B1, -1, 1}, {C1, -1, 1}] Your equations can be written in terms of the matrix $$M$$:

M = {{-beta^2 C11 - alpa^2 C55 + omega^2 p11, -alpa beta (C13 + C55), -alpa beta (e15 + e31)},
{-alpa beta (C13 + C55), -alpa^2 C33 - beta^2 C55 + omega^2 p11, -beta^2 e15 - alpa^2 e33},
{-alpa beta (e15 + e31), -beta^2 e15 - alpa^2 e33, beta^2 ep11 + alpa^2 ep33}};

Thread[M . {A1, B1, C1} == 0]
(*    your equations    *)


In general, this matrix $$M$$ has three nonzero eigenvalues and therefore $$\{0,0,0\}$$ is the only solution of these equations:

Eigenvalues[M] /. {p11 -> 1, omega -> .2, C11 -> 2.030,
beta -> 1, C55 -> 0.600, alpa -> 1,
C13 -> 0.750, e31 -> 0.200, e15 -> 3.700,
C33 -> 2.450, e33 -> 1.300, ep11 -> 38.90,
ep33 -> 25.70}
(*    {-4.74509, -1.43481, 65.1799}    *)
`

You could now ask: under what conditions does this matrix $$M$$ have a zero eigenvalue, and therefore a nontrivial solution space (nullspace)?