If A = \(\begin{bmatrix} 2 & 3 \\ 3 & 5 \end{bmatrix}\) then value of det(\(A^{2025} - 3A^{2024} + A^{2023}\)) :
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The Cayley-Hamilton theorem (\(A^2 - \text{tr}(A)A + \det(A)I = 0\) for a 2x2 matrix) is extremely useful for simplifying matrix polynomials and finding powers of matrices.
Concept:
For grazing emergence, the angle of emergence \(e = 90^\circ\). This implies the internal angle at the second face \(r_2\) is equal to the critical angle \(C\).
Relation for prism: \(A = r_1 + r_2\).
\includegraphics[width=0.5\linewidth]{19 ans.png}
Step 1: Find the critical angle.
\[ \sin C = \frac{1}{\mu} = \frac{1}{\sqrt{2}} \]
\[ C = 45^\circ \]
So, \(r_2 = 45^\circ\).
Step 2: Calculate \(r_1\).
For an equilateral prism, \(A = 60^\circ\).
\[ r_1 + r_2 = A \]
\[ r_1 + 45^\circ = 60^\circ \]
\[ r_1 = 15^\circ \]
Step 3: Conclusion.
The angle of refraction at the first surface is \(15^\circ\).
