Let $A$ be $5\times 5$ $\textit{complex}$ matrix such that $(A^2-I)^2=0$. Assume that $A$ is not a diagonal matrix. Which of the following statement is true ?
$A$ is diagonalizable.
$A$ is NOT diagonalizable.
No conclusion can be drawn about the diagonalizability of $A$.
I feel that third option is correct but I don't know how to construct such counterexamples for first both options. Can someone please give me any hint ?
The given condition is equivalent to say that $\;A\;$ is a zero of the polynomial $\;(x^2-1)^2\;$, so the minimal polynomial of $\;A\;,\;\;p_A(x)\;$ is a divisor of the above polynomial.
Remember now that $\;A\;$ is diagonalizable iff $\;p_A(x)\;$ is a product of different linear polynomials, so since $\;(x^2-1)^2=(x-1)^2(x+1)^2\;$ , it is enough to find a matrix whose minimal polynomial is not $\;(x-1)\,,\,\,(x+1)\,,\,\,\text{or}\;\;(x-1)(x+1)\;$ , for example
$$A=\begin{pmatrix} 1&1&0&0&0\\ 0&1&0&0&0\\ 0&0&-1&0&0\\ 0&0&0&1&0\\ 0&0&0&0&1\end{pmatrix}$$
is such a matrix.