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Suppose $A$ is a $4\times 5$ matrix and $b \in R^4$. Decide whether the system $Ax = b$ is consistent if $dim\ NulA = dim Nul [A|b] = 4$

My question has two parts. The first is this $dim\ Nul [A|b]$.

I know it would look like this (using 2x2 for simplicity); $A=\begin{bmatrix}a_{11} && a_{12} & | & b_1\\a_{21} && a_{22} & | & b_2\end{bmatrix}$

How do I deal with things like Rank, Row Space, Null Space, etc in this kind of matrix (also, does it have a special name I can use)? Can I simply ignore the 'line' and treat it exactly as I would a normal matrix, or are there some special properties that I need to be aware of?

Assuming that I can treat it as a regular matrix, then I am just doing it like this;

If $dim\ Nul A = dim\ Nul [A|b]= 4$ then $Rk(A) = 1$ and $Rk(A|b) = 2$. This means $b$ has a non-zero entry in a zero-row of $A$ (in RREF), and therefore the system is NOT consistent.

Is that answer both correct, and sufficiently explanatory?

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Yes, you are right: we have $rank(A|b) \ne rank(A)$ and therefore the system $Ax=b$ has no solution.

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    So $A|b$ is just a normal matrix? – 2017-02-06
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    Yes. $A|b$ has the columns of $A$ and the one additional column $b$. – 2017-02-06
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    Easy. Thank you very much for your help :) – 2017-02-06