Diagonalization and eigenvalues Let $A$ $\in M_3(\mathbb R)$ which is not a diagonal matrix. Pick out the cases when
$A$ is diagonalizable over $\mathbb R$:
a. when $A^2 = A$;
b. when $(A - 3I)^2 = 0$;
c. when $A^2 + I = 0$.
I could eliminate c. by using the equation $\lambda^2+1$ and showing if a matrix has to satisfy this then it has to be diagonal.
I am also in doubt whether all the eigen values of $A$ satisfy this or not.

Not sure what the question is, so I will take a stab at the answer. I am assuming you want to know what additional condition is needed to ensure that it is diagonalizable.
a) $A^2=A$
In this case the eigenvalues of $A$ have to be zero or one. Since we need three eigenvalues,
at least one of them is repeated. Now Jordon block of the repeated eigenvalue has to be diagonal for (a) to hold. Hence $A$ is diagonalizable.
b) $(A-3I)^2=0$
This need not be diagonalizable, unless $A-3I=0$. If this is not true then A has to be similar to 
$$A = \begin{pmatrix}3 & 1 & 0\cr 0 & 3 & 0\cr 0 & 0 & 3\end{pmatrix}$$
which can't be diagonalized.
c) A^2+I = 0
This can't be diagonalized in the field of real numbers. This is because the eigenvalues of any diagonal matrix are the values in the diagonals. However, $A$ has eigenvalues $\pm \sqrt{-1}$.
