) < A = .In other words, the homogeneous system ( ) has a non-trivial solution if and only if the determinant of the coefficient matrix is zero. Suppose that m > , then there are more number of equations than the number of unknowns. Reducing the system by elementary transformations, we get ρ ρ ( ) ([ ]) A O - - Example . Solve the following system: Here the number of equations is equal to the number of unknowns.
Transforming into echelon form (Gaussian elimination method), the augmented matrix becomes ÷ − ÷ − ) , ÷ − So, ρ ρ ( ) ([ ]) A O Number of unknowns Hence, the system has a unique solution. Since x is always a solution of the homogeneous system, the only solution is the trivial solution x Note In the above example, we find that A = = −≠ Example . Solve the system: x Here the number of unknowns is . Transforming into echelon form (Gaussian elimination method), the augmented matrix becomes ÷ − ÷ − ) , So, ρ ρ ( ) ([ ]) A O < Number of unknowns Hence, the system has a one parameter family of solutions.
Writing the equations using the echelon form, we get Taking z = , where t is an arbitrary real number, we get by back substitution, - - Applications of Matrices and Determinants ⇒ + − ⇒ ⇒ = − . So, the solution is x t y t z , where is any real number. Example . Solve the system: x Here the number of equations is and the number of unknowns is .