##### Exercise4

Our definition of an invertible matrix requires that $$A$$ be a square $$n\times n$$ matrix. Let's examine what happens when $$A$$ is not square. For instance, suppose that

\begin{equation*} A = \left[\begin{array}{rr} -1 \amp -1 \\ -2 \amp -1 \\ 3 \amp 0 \\ \end{array}\right], \hspace{24pt} B = \left[\begin{array}{rrr} -2 \amp 2 \amp 1 \\ 1 \amp -2 \amp -1 \\ \end{array}\right] \text{.} \end{equation*}
1. Verify that $$BA = I_2\text{.}$$ In this case, we say that $$B$$ is a left inverse of $$A\text{.}$$

2. If $$A$$ has a left inverse $$B\text{,}$$ we can still use it to find solutions to linear equations. If we know there is a solution to the equation $$A\xvec = \bvec\text{,}$$ we can multiply both sides of the equation by $$B$$ to find $$\xvec = B\bvec\text{.}$$

Suppose you know there is a solution to the equation $$A\xvec = \threevec{-1}{-3}{6}\text{.}$$ Use the left inverse $$B$$ to find $$\xvec$$ and verify that it is a solution.

3. Now consider the matrix

\begin{equation*} C = \left[\begin{array}{rrr} 1 \amp -1 \amp 0 \\ -2 \amp 1 \amp 0 \\ \end{array}\right] \end{equation*}

and verify that $$C$$ is also a left inverse of $$A\text{.}$$ This shows that the matrix $$A$$ may have more than one left inverse.

4. When $$A$$ is a square matrix, we said that $$BA=I$$ implies that $$AB=I\text{.}$$ In this problem, we have a non-square matrix $$A$$ with $$BA = I\text{.}$$ What happens when we try to compute $$AB\text{?}$$

in-context