# What does it mean when a system of linear equations have no solution?

$$A = \left( \begin{array}{ccc} 10 & 29 & 41 \\ 23 & 27 & 42 \\ 24 & 28 & 48 \\ \end{array} \right)$$ $\det (A) = -1748$.

Now $B$ is formed when the second column is multiplied by $15$ and added to the first column. $$B = \left( \begin{array}{ccc} 445 & 29 & 41 \\ 428 & 27 & 42 \\ 444 & 28 & 48 \\ \end{array} \right)$$ $\det (B) = -1748$.

When i reduced these two matrices the final row is $[0,0,1]$, which means there are no solutions.

Why is that? If my system is inconsistent then what does it mean?

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First of all, if your final row is $[0,0,1]$ it is not a row of "zeros". Secondly, if you have pivots (leading ones) in every column then there is a solution. In this case, since $\det(A) = \det(B) \neq 0$ the matrix is invertible and your solution is unique. Third of all, I think you meant $\det(B)$ after you define $B$, not $\det(A)$. – user139388 Apr 6 '14 at 2:01
1) Corrected. 2) I didn't get that statement. 3) Corrected as well. – Miodrag Apr 6 '14 at 2:06
What makes you think that there are no solutions when you get a final row of $[0,0,1]$? The only way for there to be no solutions is if your system is $Ax=b$ with $b \neq 0$, and such that when you reduce to the RREF of $[A \mid b ]$ you have a leading $1$ in the last column. – user139388 Apr 6 '14 at 2:12
@Amzoti No i don't have that system. I simply used my graphics calculator to get that final row using the function RREF. – Miodrag Apr 6 '14 at 2:24

In the case you've given, the matrix of coefficients $A$ has nonzero determinant, so you know (depending on what you've learned so far) that the system has a unique solution.
As far as what an inconsistent system means, consider the case of two equations in two unknowns. Each of those equations represents a line in the $xy$-plane, and a solution to the system is an intersection point of those lines. If the lines are nonparallel, they intersect in exactly one point and the system has a unique solution. If they coincide, there are an infinite number of solutions. But if they are parallel and not the same line, they do not intersect, so the system has no solutions and is inconsistent. The situation with more variables and more equations is more complicated when you try to visualize it, but is conceptually the same: if the surfaces defined have a unique intersection point, the system has a unique solution, and so on.