Trace of a Matrix

Given,

$$A=\begin{pmatrix}1&2\\-1&3\end{pmatrix}$$

First find $$A^2=A\cdot A$$

$$A^2=\begin{pmatrix}1&2\\-1&3\end{pmatrix}\begin{pmatrix}1&2\\-1&3\end{pmatrix}$$

First row first column : $$1(1)+2(-1)=1-2=-1$$

First row second column : $$1(2)+2(3)=2+6=8$$

Second row first column : $$(-1)(1)+3(-1)=-1-3=-4$$

Second row second column : $$(-1)(2)+3(3)=-2+9=7$$

Therefore, $$A^2=\begin{pmatrix}-1&8\\-4&7\end{pmatrix}$$

Now find $$4A$$

$$4A=4\begin{pmatrix}1&2\\-1&3\end{pmatrix}=\begin{pmatrix}4&8\\-4&12\end{pmatrix}$$

Now calculate $$A^2-4A$$

$$=\begin{pmatrix}-1&8\\-4&7\end{pmatrix}-\begin{pmatrix}4&8\\-4&12\end{pmatrix}$$

Subtract corresponding entries :

$$=\begin{pmatrix}-1-4&8-8\\-4-(-4)&7-12\end{pmatrix}$$

$$=\begin{pmatrix}-5&0\\0&-5\end{pmatrix}$$

Trace of a matrix is sum of diagonal elements.

Therefore,

$$\text{Trace}(A^2-4A)=-5+(-5)=-10$$

Final Answer :

$$-10$$

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:
$$\frac{3I}{\pi^3} = \frac{3}{\pi^3} \cdot \frac{\pi^3}{3} = 1$$