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.

Yet another take on this is provided in my answer to this question's evil twin, Matrix diagonalizable or not!
Hope this helps.  Cheers,
and as always,
Fiat Lux!!!
