Ononot — Consider the space of all 3 \times 3 matrices M3(\mathbb{R})

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Matriceshard

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Consider the space of all $3 \times 3$ matrices $M_3(\mathbb{R})$ . Let $S$ be the subset of matrices $A$ such that $A$ is idempotent ( $A^2 = A$ ) and $\text{tr}(A) = 1$ . What is the dimension of the variety of such matrices?