Question

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

Answer 1

In order for a matrix to be invertible, you need to have no zero columns or rows, and all rows and columns must be independent (i.e. no linear combinations of the others).(*)

So lets start with a 2 by 2 matrix with no zero columns and rows, the identity:

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

We can change each one of those 0s by a 1 (but not both! otherwise we could eliminate one row because it would be a linear combination of the other one), and the matrix will still be invertible, that gives us two more:

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

If we do the same but transposing the identity (exhanging its rows and columns) we get the other three:

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


(*) the first condition is really a consecuence of the first one.