1. Key Concept: De Moivre's Theorem for a Specific Matrix Form

This problem involves calculating powers of a special type of $2 \times 2$ matrix. For a matrix of the form: $M = \left[ {\begin{matrix} {\cos \theta } & {i\sin \theta } \\ {i\sin \theta } & {\cos \theta } \\ \end{matrix} } \right]$ Its $n$-th power, $M^n$, follows a pattern remarkably similar to De Moivre's Theorem for complex numbers. The theorem states: ${M^n} = \left[ {\begin{matrix} {\cos (n\theta)} & {i\sin (n\theta)} \\ {i\sin (n\theta)} & {\cos (n\theta)} \\ \end{matrix} } \right]$ Why is this important? This formula allows us to directly compute any positive integer power of such a matrix by simply multiplying the angle $\theta$ by the power $n$, avoiding tedious matrix multiplications. This is a significant shortcut for this specific matrix structure.

2. Applying the Theorem to Matrix A

We are given the matrix $A$: $A = \left[ {\begin{matrix} {\cos \theta } & {i\sin \theta } \\ {i\sin \theta } & {\cos \theta } \\ \end{matrix} } \right]$ And the angle $\theta = \frac{\pi}{24}$. We need to find $A^5$. Using the generalized De Moivre's Theorem for matrices with $n=5$: ${A^5} = \left[ {\begin{matrix} {\cos (5\theta)} & {i\sin (5\theta)} \\ {i\sin (5\theta)} & {\cos (5\theta)} \\ \end{matrix} } \right]$ Why this step? By recognizing the given matrix $A$ fits the specific form for which De Moivre's theorem applies, we can immediately write down the general form of $A^5$.

3. Calculate the Angle $5\theta$

Substitute the given value of $\theta$ into $5\theta$: $5\theta = 5 \times \frac{\pi}{24} = \frac{5\pi}{24}$ Why this step? This provides the specific angle required for the elements of $A^5$.

4. Determine the Elements $a, b, c, d$ of $A^5$

Now substitute $5\theta = \frac{5\pi}{24}$ into the expression for $A^5$: ${A^5} = \left[ {\begin{matrix} {\cos \left( {\frac{5\pi}{24}} \right)} & {i\sin \left( {\frac{5\pi}{24}} \right)} \\ {i\sin \left( {\frac{5\pi}{24}} \right)} & {\cos \left( {\frac{5\pi}{24}} \right)} \\ \end{matrix} } \right]$ We are given that {A^5} = \left[ {\matrix{ a & b \cr c & d \cr } } \right]. By comparing the elements, we get: $a = \cos \left( {\frac{5\pi}{24}} \right)$ $b = i\sin \left( {\frac{5\pi}{24}} \right)$ $c = i\sin \left( {\frac{5\pi}{24}} \right)$ $d = \cos \left( {\frac{5\pi}{24}} \right)$ Why this step? These are the explicit values of $a, b, c, d$ that we will use to evaluate the given options. Note that $a$ and $d$ are real, while $b$ and $c$ are purely imaginary (since $\sin(\frac{5\pi}{24}) \ne 0$). Let $X = \frac{5\pi}{24}$ for brevity in the following calculations. So, $a = \cos X$, $b = i\sin X$, $c = i\sin X$, $d = \cos X$.

5. Evaluate Each Option

We need to find which of the given statements is not true.

Option (A): $a^2 - c^2 = 1$ Substitute the expressions for $a$ and $c$: $a^2 - c^2 = (\cos X)^2 - (i\sin X)^2$ $= \cos^2 X - (i^2 \sin^2 X)$ Recall that $i^2 = -1$: $= \cos^2 X - (-\sin^2 X)$ $= \cos^2 X + \sin^2 X$ Using the fundamental trigonometric identity $\cos^2 X + \sin^2 X = 1$: $= 1$ So, the statement $a^2 - c^2 = 1$ is TRUE. Why this step? This is a direct application of the definitions of $a, c$ and the identity $\cos^2 x + \sin^2 x = 1$. It confirms that this statement holds true.

Option (B): $0 \le a^2 + b^2 \le 1$ Substitute the expressions for $a$ and $b$: $a^2 + b^2 = (\cos X)^2 + (i\sin X)^2$ $= \cos^2 X + (i^2 \sin^2 X)$ $= \cos^2 X - \sin^2 X$ Using the double-angle identity $\cos^2 X - \sin^2 X = \cos(2X)$: $= \cos(2X)$ Now, calculate $2X$: $2X = 2 \times \frac{5\pi}{24} = \frac{10\pi}{24} = \frac{5\pi}{12}$ So, $a^2 + b^2 = \cos\left(\frac{5\pi}{12}\right)$. Since $\frac{5\pi}{12}$ radians is equivalent to $75^\circ$, which lies in the first quadrant ($0 < 75^\circ < 90^\circ$), the value of $\cos(75^\circ)$ is positive and less than 1. Specifically, $\cos(75^\circ) = \frac{\sqrt{6}-\sqrt{2}}{4} \approx 0.2588$. Therefore, $0 \le \cos\left(\frac{5\pi}{12}\right) \le 1$. So, the statement $0 \le a^2 + b^2 \le 1$ is TRUE. Why this step? This involves applying definitions, trigonometric identities ($\cos(2x)$), and understanding the range of the cosine function for the calculated angle.

Option (C): $a^2 - d^2 = 0$ Substitute the expressions for $a$ and $d$: $a^2 - d^2 = (\cos X)^2 - (\cos X)^2$ $= 0$ So, the statement $a^2 - d^2 = 0$ is TRUE. Why this step? This directly follows from the fact that $a=d=\cos X$ for this matrix form.

Option (D): $a^2 - b^2 = \frac{1}{2}$ Substitute the expressions for $a$ and $b$: $a^2 - b^2 = (\cos X)^2 - (i\sin X)^2$ $= \cos^2 X - (i^2 \sin^2 X)$ $= \cos^2 X - (-\sin^2 X)$ $= \cos^2 X + \sin^2 X$ Using the fundamental trigonometric identity $\cos^2 X + \sin^2 X = 1$: $= 1$ So, our calculation shows $a^2 - b^2 = 1$. The given statement is $a^2 - b^2 = \frac{1}{2}$. Since $1 \ne \frac{1}{2}$, the statement $a^2 - b^2 = \frac{1}{2}$ is NOT TRUE. Why this step? This calculation is identical to Option (A), leading to 1. Comparing this result with the proposed value of $1/2$ allows us to determine if the statement is true or false.

6. Conclusion

From our evaluation:

• Option (A) is TRUE.
• Option (B) is TRUE.
• Option (C) is TRUE.
• Option (D) is NOT TRUE.

The question asks for the statement that is not true. Therefore, option (D) is the correct answer based on these calculations.

Tips and Common Mistakes:

• Recognize the Matrix Form: The ability to identify the specific matrix form that allows for De Moivre's theorem is crucial. Don't apply it to just any matrix.
• Careful with $i^2 = -1$: A very common mistake is to forget that $i^2 = -1$, especially when squaring terms like $i\sin\theta$. This can lead to errors, for example, mistaking $\cos^2 X - (i\sin X)^2$ for $\cos^2 X - \sin^2 X$.
• Trigonometric Identities: Be proficient with fundamental identities like $\sin^2 x + \cos^2 x = 1$ and double-angle formulas like $\cos(2x) = \cos^2 x - \sin^2 x$.
• Angle Quadrants: When evaluating inequalities involving trigonometric functions (like in Option B), remember to consider the quadrant of the angle to determine the sign and range of the function.

Summary: This problem effectively tests your knowledge of De Moivre's Theorem as applied to a specific $2 \times 2$ matrix, along with fundamental complex number properties ($i^2 = -1$) and trigonometric identities. The structured approach of first finding the elements of the matrix power and then systematically checking each option ensures accuracy.