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