Title is the question, I tried taking trace both side and got trace of $A$ is zero, now to conclude $A$ is singular, suppose $A$ is non singular, then multiplying both side by Inverse of $A$ we get $B$ is similar to $B+I$, which is impossible as similar matrices have same eigen values where as eigen values of $B+I$ is one unit more than the eigen values of $B$.

Is my solution make sense? Thanks for helping, I am using Android phone so not able to write mathematical symbols as it takes too much time to write.