Someone please help to solve this.
Let A be an nxn matrix and n is odd. Show that it is not possible for A^{2} + I = O.
And what about n is even?
For $n$ even a single example will suffice to show it is possibbe for an $n\times n$ matrix $A$ to satisfy $A^2+1=0$. Trial and error or playing with the possibilities will allow you to find a $2 \times 2$ matrix with the required property.