Let $A$ be a $2\times2$ matrix with real entries such that $A^2=0$.Find the determinant of $I+A$ where $I$ denotes the identity matrix. I proceed in this way :Note that $(I+A)A=A+A^2 \Longrightarrow (I+A)A=A$ (Since $A^2=0$).
Now taking determinant both side we get $|(I+A)A| = |A| \Longrightarrow |(I+A)|\cdot|A|=|A|\Longrightarrow |I+A|=1$.
Am I proceed in correct way? In the last line am I need to consider $|A|=0$?