Problem 3 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(A=\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta\end{array}\right]\), then for \(n \in N, A^{n}\) is equal to (A) \(\left[\begin{array}{cc}\cos ^{n} \theta & \sin ^{n} \theta \\ -\sin ^{n} \theta & \cos ^{n} \theta\end{array}\right]\) (B) \(\left[\begin{array}{cc}\cos n \theta & \sin n \theta \\ -\sin n \theta & \cos n \theta\end{array}\right]\) (C) \(\left[\begin{array}{cc}\cos n \theta & -\sin n \theta \\ \sin n \theta & \cos n \theta\end{array}\right]\) (D) \(\left[\begin{array}{cc}n \cos \theta & n \sin \theta \\ -n \sin \theta & n \cos \theta\end{array}\right]\)

Step-by-Step Solution

Answer

(B) \(\cos n \theta)\) represents \(A^n\).

Step 1: Identify the Matrix Type

Recognize that \(A\) is a rotation matrix. Rotation matrices are used to rotate vectors in the plane by a given angle \(\theta\). The matrix \(A\) itself represents a rotation by angle \(\theta\) counterclockwise.

Step 2: Compute \(A^2\) for Insight

To help anticipate the pattern, calculate \(A^2 = A \times A\). Use matrix multiplication:\[ A^2 = \left[ \begin{array}{cc} \cos \theta & \sin \theta \ -\sin \theta & \cos \theta \end{array} \right] \times \left[ \begin{array}{cc} \cos \theta & \sin \theta \ -\sin \theta & \cos \theta \end{array} \right] \]This equals:\[ \left[ \begin{array}{cc} \cos^2 \theta - \sin^2 \theta & 2 \cos \theta \sin \theta \ -2 \cos \theta \sin \theta & \cos^2 \theta - \sin^2 \theta \end{array} \right] \]Using trigonometric identities, this can be simplified to \[ \left[ \begin{array}{cc} \cos(2\theta) & \sin(2\theta) \ -\sin(2\theta) & \cos(2\theta) \end{array} \right] \]

Step 3: Generalize for Any \(n\)

From the example of calculating \(A^2\), note the relationship to the angle \(2\theta\). In general, a rotation matrix raised to the power \(n\) will be a rotation by \(n\theta\):\[ A^n = \left[ \begin{array}{cc} \cos(n\theta) & \sin(n\theta) \ -\sin(n\theta) & \cos(n\theta) \end{array} \right] \]This implies that \(A^n\) represents a rotation by \(n\theta\) rather than using powers directly on sine and cosine functions individually.

Step 4: Match with Given Options

Compare the expression from Step 3 with the provided options. Option (B) is:\[ \left[ \begin{array}{cc} \cos n \theta & \sin n \theta \ -\sin n \theta & \cos n \theta \end{array} \right] \]Clearly, this matches the result we derived in Step 3.

Key Concepts

Matrix Multiplication

When working with matrices, especially rotation matrices, understanding matrix multiplication is key. Matrix multiplication is the process of multiplying two matrices by taking the dot product of rows and columns.
For example, if you have two matrices, say matrix A and B, you calculate the element in the first row and first column of the resulting matrix by multiplying each element of the first row of A by the corresponding element of the first column of B and summing the results.
Here's a quick run-through:

• Multiply each element of the rows of the first matrix by the corresponding elements of the columns of the second matrix.
• Add the products to get the entries of the new matrix.

This method is critical when calculating powers of a matrix, like finding \(A^2\), \(A^3\), and so on. In the context of rotation matrices, the matrix multiplication reflects the combination of sequential rotations.

Trigonometric Identities

Trigonometric identities are mathematical equations that relate the trigonometric functions and are key to simplifying matrices. In this exercise, we encounter terms like \(\cos^2 \theta - \sin^2 \theta\), which can be simplified using well-known trigonometric identities.
Here are few identities that are particularly useful:

• \(\cos(2\theta) = \cos^2 \theta - \sin^2 \theta\)
• \(\sin(2\theta) = 2\sin \theta \cos \theta\)

These identities help in transforming expressions into forms that are easier to manage, such as changing \(\cos^2 \theta - \sin^2 \theta\) into \(\cos(2\theta)\). Simplification using these identities allows us to solve problems involving rotation matrices more efficiently by reducing the complexity of our computations.

Angle of Rotation

The angle of rotation relates directly to how matrices influence vectors in a plane. Rotation matrices describe movements around an origin by a specific angle \(\theta\). With each multiplication of the rotation matrix by itself, the total angle of rotation increases.
Consider a rotation matrix \(A\) that rotates a vector by \(\theta\) degrees counterclockwise. If we want to rotate that vector by \(2\theta\), \(3\theta\), or even \(n\theta\) degrees, we simply multiply the rotation matrix by itself the appropriate number of times.
The formula that emerges from repeated multiplication of a rotation matrix, \(A^n\), is actually a matrix that corresponds to a single rotation angle of \(n\theta\). Therefore:

• \(A^2\) corresponds to a \(2\theta\) rotation.
• \(A^3\) corresponds to a \(3\theta\) rotation.
• \(A^n\) corresponds to an \(n\theta\) rotation.

This concept elegantly demonstrates how linear algebra and trigonometry combine to describe complex geometric transformations succinctly.