Activity3.1.2

Let's consider the matrices

\begin{equation*} A = \left[\begin{array}{rrr} 1 \amp 0 \amp 2 \\ 2 \amp 2 \amp 1 \\ 1 \amp 1 \amp 1 \\ \end{array}\right], B = \left[\begin{array}{rrr} 1 \amp 2 \amp -4 \\ -1 \amp -1 \amp 3 \\ 0 \amp -1 \amp 2 \\ \end{array}\right] \text{.} \end{equation*}
  1. Define these matrices in Sage and verify that \(BA = I\) so that \(B=A^{-1}\text{.}\)

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

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

  4. 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 \(A\text{?}\)

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

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

in-context