What is the meaning of Rank[A | b]? (Linear Algebra)

Question

I got a question in my textbook where I am supposed to find if the linear system Ax=b is consistent. Then we are given some information. I do think I know how to solve this kind of problem but they use this notation Rank$[A | $b$]$ where it is some kind of number. What does this notation mean?

Answer 1

If the system of equations is written in matrix form, $A\mathbf{x}=\mathbf{b}$, then $A$ is the coefficient matrix and $[A|\mathbf{b}]$ is the so called augmentend matrix: the matrix formed by adding a column to $A$, consisting of the constants $\mathbf{b}$ from the system of equations (the right-hand side). You can check the Wikipedia page for some examples.

Answer 2

I'm guessing that $[A|b]$ refers to the augmented matrix formed by augmenting the column vector $b$ onto the matrix $A$. That is, it's a matrix with one extra column: $b$.

Note that the equation $Ax = b$ has a solution if and only if $b$ is in the columnspace of $A$ and $\operatorname{Rank} [A | b] = \operatorname{Rank} A$. Otherwise, $\operatorname{Rank} [A | b] = \operatorname{Rank} A + 1$.