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Show that the linear system $AX=B$ has a solution if and only if the rank of $A$ equals the rank of the extended matrix $(AB)$.

I started by saying that $AX=a_1 x_1+a_2 x_2+\cdots+a_n x_n=B$, where $a_{i}$ represent the columns of $A$ and $x_i$ represent the coefficients of the system. It then follows that $B$ is a linear combination of the columns of $A$, so that $B \in C(A)$. Now I don't know how to go on further...

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    What is the definition of the extended (often also called augmented) matrix? In other words, do you understand how that matrix "$(AB)$" looks? Your reasoning looks good up to a certain point... until "$AB=a_{1}b_{1}+...+a_{n}b_{n}$", which doesn't make sense. Nor does "$AB \in C(A)$". – 2017-01-22
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    I guess I didn't really understand the concept of augmented matrix, but I do now, thanks for making the remark! – 2017-01-22

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