Originally Posted by

**rebghb** Hello everyone!

I'm stuck on this linear quetion that is supposed to be easy: Show that no 3x3 matrix $\displaystyle A$exists such that $\displaystyle A^2+I=0$.

Since the question is asking for a 3x3 matrix I figured it needed the long and boring way, so I considered a matrix and initialized its entries to $\displaystyle [A_1 A_2 A_3]$ where $\displaystyle A_1=[a \, x \, u]^t$, $\displaystyle A_2=[b \, y \, v]^t$, and $\displaystyle A_3=[c \, z \, w]^t$, carried out matrix product... LOST!

2nd way: $\displaystyle A^2+I=0,A^2+AA^{-1}=0 \,\text{ so }A+A^{-1}=0$. But what then??