we have a system of linear equations as such: x + 2 y + (...
we have a system of linear equations as such:
and i have to find the solution in and so i have no problem for i get the matrix
but the questions i have are as follows:
1. Can i use what i found for the augmented matrix and the discussion by parameter a in to deduce ?
2. Or is there some other way i must reduce to row echelon form for and then have the discussion for parameter a?
3. If i had an 3x3 or 4x4 system to solve over a low prime and (eg 5 and 7) how would i go about doing it with the matrix gauss elimination?could i use the same augmented matrix and reduce it to row echelon over and then use that augmented matrix for the rest like above or not?
4. If i recall correctly there was a theorem about the rank of the original matrix and augmented that says something about the number of solutions but i do not recall how that would help me find solutions just eliminate the a's where there is none?
Answer & Explanation
If in producing the echelon form you only used multiplication or division by integers, provided you never divided by a multiple of 5, the same steps would produce the echelon form over .
Assuming this is the case, the system over has solution if and only if the last column is not a pivot one. We need to distinguish the cases and . If , you can go on with Gaussian elimination to
and the last column is a pivot column.
If , the matrix is
and the system has solutions, with being a free variable (so five distinct solutions).