###### Activity 3.1.2

Let's consider the matrices

Define these matrices in Sage and verify that \(BA = I\) so that \(B=A^{-1}\text{.}\)

Find the solution to the equation \(A\xvec = \threevec{4}{-1}{4}\) using \(A^{-1}\text{.}\)

Using your Sage cell above, multiply \(A\) and \(B\) in the opposite order; that is, what do you find when you evaluate \(AB\text{?}\)

Suppose that \(A\) is an \(n\times n\) invertible matrix with inverse \(A^{-1}\text{.}\) This means that every equation of the form \(A\xvec=\bvec\) has a solution, namely, \(\xvec = A^{-1}\bvec\text{.}\) What can you conclude about the span of the columns of \(A\text{?}\)

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

If \(A\) is an invertible \(4\times4\) matrix, what is its reduced row echelon form?