$A\in M_3(\mathbb R)$ orthogonal matrix with $\det (A)=1$.
I need to prove that $(\mathrm{tr} A)^2-\mathrm{tr}(A^2) = 2 \mathrm{tr} (A)$ ; $\mathrm{tr}$=trace.
I know that if $A$ is orthogonal than $A^tA=I$ and that $A$ is diagonalizable and similar to $D=\begin{pmatrix} \lambda_1 & & \\ & \lambda_2 & \\ & & \lambda_3 \end{pmatrix}$. We know as well that $\mathrm{tr} A=\mathrm{tr} D= \lambda_1+ \lambda_2+\lambda_3$ that $\mathrm{tr} A^2=\mathrm{tr} D^2= \lambda_1^2+ \lambda_2^2+\lambda_3^2$ and that $\lambda_1 \lambda_2 \lambda_3=1$. It's not enough for solving the question.
What more should I know or use in order to solve it?
Thanks