We can determine whether or not a linear system $Ax = b$ has a unique solution or a solution at all based on the rank of the augmented matrix rank$(A|b)$ and the rank of the matrix rank$(A)$. (e.g.: Linear system solutions)
What happens in a more complex linear system where $b$ is replaced by $B$ which has multiple columns? For example:
Be $A \in K^{i,j}$ and $B \in K^{i,k}$ with $i,j,k \in \mathbb{N}$. When does the linear system $AX = B$ have at least one solution $\hat{X} \in K^{j,k}$?
Best Answer
Let $X = (x_1,\ldots,x_k)$ and $B = (b_1,\ldots,b_k)$. Then $AX = B$ is equivalent to $Ax_n = b_n$ for all $n \in \{1,\ldots,k\}$. Now you can apply your theorem recursively to obtain that $AX = B$ has a solution if and only if $\mathrm{rank}(A|B) = \mathrm{rank}(A)$.