Answer:
Step-by-step explanation:
If the columns span R^6, this says that every b in R^6 is in the span of the columns, which is another way of saying that any b is a linear combination of the columns. Then the equation is consistent for some b or all b in R^6.
If an augmented matrix can be transformed by elementary row operations into reduced echelon form, then the equation Ax = b is consistent.
Any matrix can be transformed by elementary row operations into reduced echelon form, but not every matrix equation Ax = b is consistent
For example, after echelon reduction we have
1 6. 7. 8. 8.
0 1. 5. 4. 9
0. 0. 1. 3. 6
0. 0. 0. 1. 6.
0. 0. 0. 0. 1
0. 0. 0. 0. 0
In this case, the matrix doesn't have a solution, then the matrix is inconsistent
But if the matrix echelon reduction gives
1 6. 7. 8. 8.
0 1. 5. 4. 9
0. 0. 1. 3. 6
0. 0. 0. 1. 6.
0. 0. 0. 0. 1
0. 0. 0. 0. 1
Then, the system have a solution but not unique, the system have a solution for some b in R^6.
Then, the solution is II and III
So the correct option is D.
