G(n) is the number of n x n matrices J with real entries that satisfy J^2 + In = 0. Show that G(n)=0 iff n is odd.

Question

Since In is the identity matrix of order n, visualizing $J^2=-In$, how can I show that J can be a real matrix only when n is even?

Answer 1

$|J|^2 = |-I_n| = (-1)^n$

If n is odd, then $|J|^2 = -1$, not possible for a real matrix.

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btw, to make the proof complete, you still need to prove that when n is even, you can always construct at least one such $J$.

For n=2, one example would be $$ \left[ \begin{aligned} 0 && 1 \\ -1 && 0 \end{aligned}\right] $$

but you still need to find a general form for all odd n.