Medium

Probability

Suppose you start with the $2 \times 2$ identity matrix $I_2$. At each step, select one of the four elements of the matrix uniformly at random. If the element you select is a $1$, change it to $0$. If the element you select is a $0$, change it to $1$. Find the expected number of steps needed to obtain a singular matrix (i.e. the determinant is 0).

Notes

Hint