# At least one solution to equation

Given edit: an equation in matrix form $$\underline{\underline{A}} \ \underline{x} = \underline{B}$$ and A is known, what must be true for $\underline{B}$ for the equation to have at least one solution?

I'm not quite sure if the following is sufficient, but I put the matrix $(A \ B )$ in reduced row echeleon form, and then simply noted the $b_is$ in the last column and made sure there wasn't a pivot.

Is that the correct method, or am I missing something? I got $$\begin{pmatrix} 1 & 1 & 0 & b_1 - b_2/2 \\ 0 & 0 & 1 & b_2/2 \\ 0 & 0 & 0 & b_3 - b_1 + b_2\end{pmatrix}$$ Would the argument then be that the entry in 4th row, 4th column must equal zero?

• What is the underlining and double underlining supposed to signify? – Zubin Mukerjee Nov 29 '14 at 22:35
• Matrices, all of them (the X and B) are column matrices, that's why I only put one line. Sorry if this wasn't commonly known, of course that might not be true internationally... – John Dane Nov 29 '14 at 22:36

Let $A$ a matrix of $m$ rows and $n$ columns and $B$ a column vector of $m$ elements. Then the system of equations $Ax=B$ has an unique solution if and only if satisfies these two conditions: $r=rank(A|B)=rank(A)$ and $r=n$