Problem 3 - A Complete Resource Book in Mathematics for JEE Main 2017

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

Option (C) is correct: rotation matrix for \( n\theta \).

Step 1: Identify the Matrix and Its Property

The given matrix \( A \) is a rotation matrix in two dimensions,\[ A = \begin{bmatrix} \cos \theta & \sin \theta \ -\sin \theta & \cos \theta \end{bmatrix} \]This represents a rotation by angle \( \theta \). It has a property that its powers represent rotation by multiples of the angle \( \theta \).

Step 2: Explore the Matrix Multiplication Result

When you multiply a rotation matrix by itself, the result is another rotation matrix rotated by an additional \( \theta \). In general,\[ A^n = \begin{bmatrix} \cos n\theta & \sin n\theta \ -\sin n\theta & \cos n\theta \end{bmatrix} \]This behavior is due to the nature of how rotations compose: rotating twice by \( \theta \) yields a rotation by \( 2\theta \), and so forth.

Step 3: Analysis of Options A through D

- Option (A) reflects each element being raised to the \( n^{th} \) power, which doesn't correspond to matrix multiplication results.- Option (B) correctly places \( \cos n\theta \) and \( \sin n\theta \) in both columns, but lacks the proper signs for a rotation matrix.- Option (C) correctly depicts the rotation matrix after power \( n \) with correct sign convention.- Option (D) spreads the factor \( n \), which isn't how powers of rotation matrices work.

Step 4: Select the Correct Option

Option (C), \[ \begin{bmatrix} \cos n\theta & -\sin n\theta \ \sin n\theta & \cos n\theta \end{bmatrix} \], correctly represents the matrix raised to the \( n^{th} \) power as it reflects the nature of rotation matrices and their powers.

Key Concepts

Matrix Multiplication

Matrix multiplication is central to understanding rotation matrices. When you multiply two matrices, you're essentially composing two transformations. In the context of rotation matrices, it's like adding up the angles of rotation.
For example, if you have a rotation matrix representing a rotation by angle \( \theta \), and you multiply it by itself, the result is a rotation by \( 2\theta \). This is because the composition of two rotations results in a single rotation equal to the sum of the two angles. Therefore, for a matrix \( A \), given by:
\[ A = \begin{bmatrix} \cos \theta & \sin \theta \ -\sin \theta & \cos \theta \end{bmatrix} \]
When you calculate \( A^n \), you're effectively creating a new matrix that represents a rotation by \( n\theta \). This illustrates the cumulative effect of continuing to "rotate."
Remember, matrix multiplication is not commutative: the order in which you multiply matrices matters! This is a very important characteristic to keep in mind, especially as you delve deeper into linear algebra concepts.

Angle of Rotation

The angle of rotation \( \theta \) in a rotation matrix indicates how much to rotate a point in the plane. For a matrix:
\[ A = \begin{bmatrix} \cos \theta & \sin \theta \ -\sin \theta & \cos \theta \end{bmatrix} \]
"\( \theta \)" is the angle of rotation, and every power of this matrix, \( A^n \), rotates by angle \( n\theta \). Each multiplication effectively doubles the rotational effect.
Visualize it like this: if your original angle rotates an object 30 degrees, then the square of the matrix would rotate it 60 degrees, the cube would rotate it 90 degrees, and so on.
This sequential addition of the angle is what transforms simple rotations into powerful navigational tools in mathematics and physics. Knowing how to manipulate the angle \( \theta \) helps you understand how rotations interact in higher dimensions as well.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, play a vital role in describing rotations. They determine how far along the x and y axes a point moves during a rotation.
The general form of a 2D rotation matrix is:
\[ \begin{bmatrix} \cos \theta & \sin \theta \ -\sin \theta & \cos \theta \end{bmatrix} \]
Here, \( \cos \theta \) and \( \sin \theta \) give x and y components of a rotated vector, respectively.
These trigonometric functions change cyclically with the angle \( \theta \), allowing for a smooth transition as you rotate through 360 degrees.

• The cosine function gives the horizontal displacement.
• The sine function represents the vertical displacement.

By effective use of these functions in matrix form, we convert the simple mathematical concepts into versatile geometric transformations that are foundational to graphics, simulations, and robotics.