Reducing the augmented matrix to echelon-form, we get ] A O ÷ − ÷ − ) , ÷ − ÷ ) , So, Number of unknowns Hence the system ρ ρ ( ) ([ ]) A O has only the trivial solution Example . Determine the values of λ for which the following system of equations has a non-trivial solution. Here the number of unknowns is . So, if the system is consistent and has a non-trivial solution, then the rank of the coefficient matrix is equal to the rank of the augmented matrix and is less than .
So the determinant of the coefficient matrix should be . Hence we get = or (by applying R or ( = (by taking out ( λ − from R ) or ( = (by applying R R R or ( )( = . So λ = and λ = . We now give an application of system of linear homogeneous equations to chemistry.
You are already aware of balancing chemical reaction equations by inspecting the number of atoms present on both sides. A direct method is explained as given below. Example . By using Gaussian elimination method, balance the chemical reaction equation: C H O CO H O (The above is the reaction that is taking place in the burning of organic compound called isoprene.) We are searching for positive integers x x such that x C H x O x CO x H O ..
( ) The number of carbon atoms on the left-hand side of ( ) should be equal to the number of carbon atoms on the right-hand side of ( ). So we get a linear homogenous equation ⇒ ... ( ) Similarly, considering hydrogen and oxygen atoms, we get respectively, ⇒ = ... ( ) ⇒ = .