. . Homogeneous system of linear equations We recall that a homogeneous system of linear equations is given by a x a x a x a x a x a x a x a x a x m m m mn ... ( ) where the coefficients a i m j ij , , , ; , , are constants.
The above system is always satisfied by x x n This solution is called the trivial solution of ( ). In other words, the system ( ) always possesses a solution. The above system ( ) can be put in the matrix form AX O m , where X m m m mn O m We will denote O m × simply by the capital letter O .Since O is the zero column matrix, it is always true that ρ ρ ( ) ([ ]) A O m So, by Rouché - Capelli Theorem, any system of homogeneous linear equations is always consistent. Suppose that m < ,then there are more number of unknowns than the number of equations.
So ρ ρ ( ) ([ ]) A O < Hence, system ( ) possesses a non-trivial solution. Suppose that m = , then there are equal number of equations and unknowns: a x a x a x a x a x a x a x a x a x a x a x a x nn , ... ( ) Two cases arise. Case (i) If ρ ρ ( ) ([ ]) A O then system ( ) has a unique solution and it is the trivial solution.
Since ρ ( ) A ¹ . So for trivial solution | A ¹ . Case (ii) If ρ ρ ( ) ([ ]) A O < then system ( ) has a non-trivial solution . Since ρ (