# Changing $b$ of $AX =b$ [closed]

Suppose 3 planes of the form $$AX=b$$ intersect at a unique point. Suppose in this case $$b= (1,1,1)$$. In case $$b$$ is changed to $$(2,3,4)$$ or any other vector, will the planes still intersect at a unique point?

## closed as off-topic by GNUSupporter 8964民主女神 地下教會, Nosrati, Davide Giraudo, Namaste, KReiserDec 8 '18 at 0:21

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• Is $A$ a vector or a matrix? As written, each row of $A$ corresponds to the coefficients of a plane. What are your thoughts? – Michael Burr Dec 7 '18 at 12:01

## 1 Answer

In the first case the rank of $$[A \ b]$$ is $$3$$ (unique solution) and the system is consistent. So rank of $$A$$ is $$3$$. So any point/vector $$b\in\mathbb {R}^3$$ can be expressed as a linear combination of columns of $$A$$. Equivalently there is a unique solution.