[TIFR-GS 2011, Part B, problem no. 1] Let $A$ be a $2×2$ matrix with complex entries. The no. of $2×2$ matrices A with complex entries satisfying the equation $A^3=A$ is infinite. They asked if the statement is true or false.
From the given equation, I've found that the eigenvalues of $A$ can be $0$, $1$, and $-1$. Now I took a trial matrix and tried to satisfy the equation, but it doesn't help in any way. Any help would be appreciated.
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Since $X^3-X$ has only roots with multiplicity $1$, such a matrix $A$ has to be diagonalizable. So you are looking at the set $$\left\{A=P\left(\matrix{a&0\\0&b}\right)P^{-1}\mid P\text{ is invertible and }a,b\in\{-1,0,1\}\right\}$$ The three subsets corresponding to $a=b$ have one element each, now what happens when $a\neq b$?
For instance, $$\left\{\left(\matrix{1&0\\z&1}\right)\left(\matrix{1&0\\0&-1}\right)\left(\matrix{1&0\\-z&1}\right)\mid z\in\Bbb C\right\}$$ is infinite.
For any basis $\{x_1,x_2\}$ of $\mathbb{C}^2$, any linear transformation $A:\mathbb{C}^2\to\mathbb{C}^2$ with eigenvectors $x_1$ and $x_2$ and eigenvalues $-1, 0,$ or $1$ satisfies the equation.