# Are the invariants related in the characteristic equation of an orthogonal matrix (3x3 matrix)?

The characteristic equation of matrix A is $$\lambda ^3 - I_1\lambda^2 + I_2\lambda-I_3 = 0$$

For orthogonal matrix $$I_3 = det(A) = \pm1$$ $$I_1 = tr(A)$$

Taking examples of orthogonal matrices, it looks like $$I_1 = I_2$$. Is this true always for an orthogonal matrix? Is there some proof?

This isn't true. E.g. when $$A$$ is the negative of the identity matrix, its characteristic polynomial is $$(x+1)^3=x^3+3x^2+3x+1$$.
In general, the characteristic polynomial of a $$3\times3$$ nonsingular matrix $$A$$ is in the form of \begin{aligned} &x^3-\sum_i\lambda_i(A)x^2+\sum_{i\ne j}\lambda_i(A)\lambda_j(A)x-\prod_i\lambda_i(A)\\ =\,&x^3-\operatorname{tr}(A)x^2+\det(A)\operatorname{tr}(A^{-1})x-\det(A). \end{aligned} You are essentially asking whether $$\operatorname{tr}(A)=\det(A)\operatorname{tr}(A^{-1})$$ when $$A$$ is orthogonal. Since $$A^{-1}=A^T$$ in this case, the above equality is true if and only if $$\operatorname{tr}(A)=0$$ or $$\det(A)=1$$.