... ( ) Equations ( ), ( ), and ( ) constitute a homogeneous system of linear equations in four unknowns. The augmented matrix is [ ] A B = By Gaussian elimination method, we get ] A B ↔ ↔ Therefore, Number of unknowns ρ ρ ( ) ([ ]) A B < The system is consistent and has infinite number of solutions. Writing the equations using the echelon form, we get So, one of the unknowns should be chosen arbitrarily as a non-zero real number.
Let us choose x t t ≠ .Then, by back substitution, we get x t x t x Applications of Matrices and Determinants Since x x , and are positive integers, let us choose t = . Then, we get x = and = . So, the balanced equation is C H O CO H O Example . If the system of equations px by cz ax qy cz ax by rz has a non-trivial solution and p a q b r ¹ ¹ ¹ , prove that q q = .
Assume that the system px by cz ax qy cz ax by rz has a non-trivial solution. So, we have q = . Applying R in the above equation, we get q q That is, Since p a q b r ¹ ¹ ¹ , we get ( )( )( a q b r q . So, we have q .
Expanding the determinant, we get q = . That is, q q q q q ⇒