Random bit flips on the 2x2 identity until singular

Begin with the $2\times 2$ identity matrix. At each step, one of its four entries is chosen uniformly at random and toggled (a $1$ becomes $0$ and a $0$ becomes $1$). What is the expected number of steps until the matrix first has determinant $0$?

$\tfrac{12}{7}$. From $I$, toggling a diagonal entry makes it singular at once; toggling an off-diagonal entry gives a unit triangular matrix such as $\left(\begin{smallmatrix}1&1\\0&1\end{smallmatrix}\right)$. From there three of the four toggles produce a singular matrix and one returns to $I$. So $E_I=1+\tfrac12E_T$ and $E_T=1+\tfrac14E_I$, giving $E_I=\tfrac{12}{7}$.