Exercise 4

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 $$A = \left[\begin{array}{rr} -1 & -1 \\ -2 & -1 \\ 3 & 0 \\ \end{array}\right], \hspace{24pt} B = \left[\begin{array}{rrr} -2 & 2 & 1 \\ 1 & -2 & -1 \\ \end{array}\right]$$.

1. Verify that $BA = I_2$. In this case, we say that $B$ is a left inverse of $A$.

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

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

3. Now consider the matrix $$C = \left[\begin{array}{rrr} 1 & -1 & 0 \\ -2 & 1 & 0 \\ \end{array}\right]$$ and verify that $C$ is also a left inverse of $A$. This shows that the matrix $A$ may have more than one left inverse.