Orthogonal Matrix

We use the determinant rules for orthogonal matrices to simplify the integrand before integration.

1. The condition $$A^T = A^{-1}$$ defines an orthogonal matrix. Taking the determinant of both sides gives:
$$\det(A^T) = \det(A^{-1}) \implies \det(A) = \frac{1}{\det(A)} \implies (\det(A))^2 = 1$$
This means $$\det(A) = 1$$ or $$\det(A) = -1$$.

2. For a real $$2 \times 2$$ matrix, the trace is the sum of its eigenvalues, and the determinant is the product of its eigenvalues. An orthogonal matrix with a trace of 1 corresponds to a rotation matrix with an angle of $$\frac{\pi}{3}$$. The determinant of a rotation matrix is always positive, so $$\det(A) = 1$$.

3. For a $$2 \times 2$$ matrix, scaling by a variable factor $$x$$ modifies the determinant according to the polynomial scaling rule:
$$\det(x A) = x^2 \det(A)$$

4. Substitute this back into the definite integral:
$$I = \int_{0}^{\pi} x^2 (1) \, dx = \int_{0}^{\pi} x^2 \, dx$$

5. Integrate using the power rule:
$$I = \left[ \frac{x^3}{3} \right]_{0}^{\pi} = \frac{\pi^3}{3} - 0 = \frac{\pi^3}{3}$$

6. Calculate the final requested value:
$$\frac{3I}{\pi^3} = \frac{3}{\pi^3} \cdot \frac{\pi^3}{3} = 1$$