# If $A^2=-I$, Prove that $\det{A}=1$

If $A^2=-I$ , where $A$ is a square matrix of order $n$ and which contains real entries only and $I$ is identity matrix. Then how can we prove that $\det(A)=1$?.

I could prove that $n$ should be an even integer. But could not proceed to prove that $\det(A)$ can take only $1$, finding out few matrices which satisfies such properies (of small order) also verifies the given statement that the determinant is only $1$ and not $-1$.

Can anyone help with a hint ?

• I assume $n$ is even? Sep 25, 2013 at 15:28
• $\det(A^2)=\det(A)^2=\det(-I)=-1$, so maybe your question's wrong Sep 25, 2013 at 15:30
• @Shuchang: $\det(-I)=(-1)^n.$ Hence user7530's comment. Sep 25, 2013 at 15:32
• @user7530: Why did u remove the post? Sep 25, 2013 at 15:33
• @CameronBuie Oh, what was I thinking. Thanks Sep 25, 2013 at 15:39

Let $\lambda$ be an eigenvalue of $A$, with eigenvector $v$.
Then $$v^HA^2v = v^H(-I)v.$$ What can you conclude about $\lambda$?
• Got it . The eigenvalues are $i$ and $-i$ and the determinant is $1$.Thanks Sep 25, 2013 at 15:41