Question

Suppose \(A\) is \(m \times n\)and \(b\) is in \({\mathbb{R}^m}\). What has to be true about the two numbers rank \(\left[ {A\,\,\,{\rm{b}}} \right]\) and \({\rm{rank}}\,A\) in order for the equation \(Ax = b\) to be consistent?

Short answer

The condition \({\rm{rank}}\,\left[ {A\,\,\,{\rm{b}}} \right] = {\rm{rank}}\,A\) must be true about the two numbers rank \(\left[ {A\,\,\,{\rm{b}}} \right]\) and \({\rm{rank}}\,A\) in order for the equation \(Ax = b\) to be consistent.

Step-by-step solution

Assume an arbitrary system and relate the given statement with it

Consider the nonhomogeneous system \(Ax = b\) with the joint matrix \(\left[ {A\,\,\,{\rm{b}}} \right]\). If \(b\) is not a pivot column of the matrix \(\left[ {A\,\,\,{\rm{b}}} \right]\), then the rank of matrix \(A\) and the joint matrix \(\left[ {A\,\,\,{\rm{b}}} \right]\) are the same, that is, \({\rm{rank}}\,\left[ {A\,\,\,{\rm{b}}} \right] = {\rm{rank}}\,A\).

Use the existence and uniqueness theorem

By the existence and uniqueness theorem, a linear system is consistent if and only if the rightmost column of the augmented matrix is not a pivot. So, for the system to be consistent, \(b\) must not be a pivot column of the matrix \(\left[ {A\,\,\,{\rm{b}}} \right]\).

Draw a conclusion

If \(b\) is not a pivot column of the matrix \(\left[ {A\,\,\,{\rm{b}}} \right]\), then \({\rm{rank}}\,\left[ {A\,\,\,{\rm{b}}} \right] = {\rm{rank}}\,A\). So, the condition \({\rm{rank}}\,\left[ {A\,\,\,{\rm{b}}} \right] = {\rm{rank}}\,A\) is true about the two numbers rank \(\left[ {A\,\,\,{\rm{b}}} \right]\) and \({\rm{rank}}\,A\) in order for the equation \(Ax = b\) to be consistent.