###### Exercise 9

Suppose that \(A\) is a \(12\times12\) matrix and that, for some vector \(\bvec\text{,}\) the equation \(A\xvec=\bvec\) has a unique solution.

What can you say about the pivot positions of \(A\text{?}\)

What can you say about the span of the columns of \(A\text{?}\)

If \(\cvec\) is some other vector in \(\real^{12}\text{,}\) what can you conclude about the equation \(A\xvec = \cvec\text{?}\)

What can you about the solution space to the equation \(A\xvec =\zerovec\text{?}\)