Dtermine how many solutions and how many parameters are poss

Dtermine how many solutions (and how many parameters) are possible for a homogeneous system of four linear equations in six variables with augmented matrix A. Assume that A has nonzero entries. Give all possibilities.

a) Rank A=2

Solution

A homogeneous system of linear equations has all the numbers on the right hand side are equal to 0. If B is the coefficient matrix of the system, then BX= 0 is the matrix representation of the system, where X = ( x₁ , x₂ , x₃ , …, x_n )^T , if the system has n variables. The homogenous system Ax = 0 always has a solution x = 0. i.e. the trivial solution. If we have a homogeneous system of four linear equations in six variables, then the matrix representation of the system is BX = 0 where B is a 4x6 matrix. Also, the number of leading variables is min (4, 6) i.e. 4 and the number of non-zero rows in the RREF of B (i.e. the rank of B) is equal to the number of leading variables. Further, the number of free variables plus the number of leading variables = 6. The homogenous system BX = 0 has non-trivial solutions if and only if there are free variables. Since the rank of B 4, there will be free variables and hence the given system will always have non-trivial solutions. If rank of B = 4, there will be 2 parameters: if rank of B = 3, there will be 3 parameters; if rank of B = 2, there will be 4 parameters and if rank of B =1, there will be 5 parameters