Let A be a 14 x 17 matrix which is in reduced row echelon form. If the general solution of the linear system Ax = 0 involves exactly 8 free variables, how many rows of A consist entirely of zeros?
I'm confused as to why my approach to this question is incorrect. My approach: There are 14 rows and 17 columns; therefore 14 equations and 16 unknowns in the homogenous linear system, which is always consistent. If there are 14 rows, that means that the maximum number of leading 1's in the RREF is 14. If there are 8 free variables, there are 8 leading variables (16-8) and therefore 6 rows remaining to be filled with entirely zeros. However, the answer is 5 rows.
Where am I going wrong? Any help would be appreciated!
If the system is $A{\bf x}=0$ then the $A$ represents the coefficients on the left hand side (not the left hand side and right hand side combined). So the system has $17$ variables. Replace $16$ by $17$ and go through your argument again.