Question

What are the six invertible 2X2 matrices whose entries are 1's and 0's? (Prob. 21 in Sect 2.5)

Answer 1

\[\left[\begin{matrix}0 & 1\\ 1& 0\end{matrix}\right] \left[\begin{matrix}0 & 1\\ 1& 1\end{matrix}\right] \left[\begin{matrix}1 & 0\\ 1& 1\end{matrix}\right] \left[\begin{matrix}1 & 1\\ 0& 1\end{matrix}\right] \left[\begin{matrix}1 & 0\\ 0& 1\end{matrix}\right] \left[\begin{matrix}1 & 1\\ 1& 0\end{matrix}\right]\]

We found these by building up examples and avoiding a row or column of all zeroes or a repeated row or column. (Making the detrminant zero.)

Answer 2

There are only 8 such matrices in total. So, a simpler way is just to list them all and eliminate the singular matrices. Namely, all zeros and all ones. The rest are fine.