Question

• Write down an augmented matrix in reduced form corresponding to a system with 3 equations and 5 variables which has infinitely many solutions and 2 free variables.

• Write down an augmented matrix in reduced form corresponding to a system with 4 equations and 5 variables which has no solutions and 2 free variables.

Answer 1

1. Let A be the augmented matrix of a system with 3 equations and 5 variables which has infinitely many solutions and 2 free variables. Then A is a 3 X 6 matrix.

Let the RREF of A be

1	0	0	-1	-2	3
0	1	0	-2	-1	4
0	0	1	-3	-1	5

Now, if X = (x,y,z,w,u)^T, then the system AX = b , where b is a non-zero vector , is equivalent to x-w-2u = 3 or, x = 3+w+2u, y-2w-u = 4 or, y = 4+2w+u and z-3w-u = 5 or, z = 5+3w+u.

Then , X = (3+w+2u,4+2w+u,5+3w+u,w,u)^T = (3,4,5,0,0)^T +w(1,2,3,1,0)^T+u(2,1,1,0,1)^T.

As may be observed, the linear system AX = b is a system with 3 equations and 5 variables which has infinitely many solutions and 2 free variables.

2. Let A be the augmented matrix of a system with 4 equations and 5 variables which has no solutions. Such a system cannot have any free variables as it is inconsistent. A system can have free variables only if it is consistent and has infinitely many solutions.