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Let $A$ belongs to $M_3(\mathbb R)$ 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 $A^2 + I = 0$

(a) is certainly true as eigen values are distinct. (c) is certainly not true as eigen values are imaginary not real. but i am not sure about (b). can anybody help? thanks.

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For (b) the matrix will not be diagonalizable unless it is precisely $3I$. What can you say about the minimal polynomial?

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