Hw12 Problem 4

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Problem: (+) Consider the solution of the equation $X^2 = I_3$ in the ring $M_3(\mathbb{R})$.

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Solution: We can rewrite this equation as $X^2 - I_3 = 0$ and then factor it to be $(X - I_3)(X+I_3) = 0$ (since $(I_3)^2 = I_3$). Two obvious solutions to this equation are $I_3$ and $-I_3$. However, since we are in the ring $M_3(\mathbb{R})$, there are more solutions due to the fact $M_3(\mathbb{R})$ has many divisors of 0.