Problem 14 - Algebra for JEE ADVANCED

Question

If \(A=\left(a_{i j}\right)_{n \times n}\) and \(f\) is a function, we define \(f(A)=\left(f\left(a_{i j}\right)\right)_{n \times n^{\circ}}\) Let \(A=\left(\begin{array}{cc}\pi / 2-\theta & \theta \\ -\theta & \pi / 2-\theta\end{array}\right) .\) Then a. \(\sin A\) is invertible b. \(\sin A=\cos A\) c. \(\sin A\) is orthogonal d. \(\sin (2 A)=2 \sin A \cos A\)

Step-by-Step Solution

Answer

a, c, and d are correct. \(\sin A\) is invertible, orthogonal, and \(\sin(2A) = 2\sin A\cos A\), but \(\sin A \neq \cos A\).

Step 1: Define the Matrix Function

Given the matrix \(A = \begin{pmatrix} \frac{\pi}{2} - \theta & \theta \ -\theta & \frac{\pi}{2} - \theta \end{pmatrix}\), we need to apply the function \(f\) element-wise to each entry in \(A\). For this problem, \(f(x) = \sin(x)\), so \(\sin(A) = \begin{pmatrix} \sin(\frac{\pi}{2} - \theta) & \sin(\theta) \ \sin(-\theta) & \sin(\frac{\pi}{2} - \theta) \end{pmatrix}\).

Step 2: Calculate Each Element of \(\sin A\)

Using trigonometric identities, calculate the sine of each element:- \(\sin\left(\frac{\pi}{2} - \theta\right) = \cos(\theta)\)- \(\sin(\theta) = \sin(\theta)\)- \(\sin(-\theta) = -\sin(\theta)\)Thus, \(\sin(A) = \begin{pmatrix} \cos(\theta) & \sin(\theta) \ -\sin(\theta) & \cos(\theta) \end{pmatrix}\).

Step 3: Analyze \(\sin A\) Properties

Compare \(\sin A\) to the usual representations and check properties:1. A matrix is invertible if its determinant is non-zero.2. A matrix is orthogonal if \(A^T A = I\).3. For \(\sin A = \cos A\), compare to \(\cos(A)\).4. Analyze \(\sin(2A)\) using double angle identity.

Step 4: Check Invertibility

Calculate the determinant: \[\det(\sin A) = \cos^2(\theta) + \sin^2(\theta) = 1\]. Since the determinant is non-zero, \(\sin A\) is invertible.

Step 5: Check Orthogonality

Verify if \((\sin A)^T(\sin A) = I\):\[(\sin A)^T = \begin{pmatrix} \cos(\theta) & -\sin(\theta) \ \sin(\theta) & \cos(\theta) \end{pmatrix}\].Multiply to find:\[\begin{pmatrix} \cos(\theta) & \sin(\theta) \ -\sin(\theta) & \cos(\theta) \end{pmatrix}\begin{pmatrix} \cos(\theta) & -\sin(\theta) \ \sin(\theta) & \cos(\theta) \end{pmatrix} = I\].Thus, \(\sin A\) is orthogonal.

Step 6: Comparison with \(\cos A\)

For \(\cos A\), using \(A = \begin{pmatrix} \frac{\pi}{2} - \theta & \theta \ -\theta & \frac{\pi}{2} - \theta \end{pmatrix}\), \(\cos\left(\frac{\pi}{2} - \theta\right) = \sin(\theta)\), leading to \[\cos A = \begin{pmatrix} \sin(\theta) & \cos(\theta) \ -\cos(\theta) & \sin(\theta) \end{pmatrix}\].Compare with \(\sin A\) and conclude \(\sin A eq \cos A\).

Step 7: Investigating \(\sin(2A)\)

Using the identity \(\sin(2x) = 2 \sin(x) \cos(x)\), find if there's consistency:\(\sin(2A) \equiv 2 \begin{pmatrix} \cos\theta & \sin\theta \ -\sin\theta & \cos\theta \end{pmatrix} \begin{pmatrix} \sin\theta & \cos\theta \ -\cos\theta & \sin\theta \end{pmatrix}\) which simplifies to identity multiplied:\[2 \sin A \cos A\]. So, identity holds true for \(\sin(2A) = 2\sin A\cos A\).

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental tools in mathematics that help simplify expressions and solve equations involving trigonometric functions like sine and cosine. These identities express relationships between different trigonometric functions and angles.
In the context of matrices, such as the problem where we solve using \( \sin(A) \), these identities become handy when determining the sine of angles that appear as elements in a matrix. For instance, using the trigonometric identity \( \sin\left(\frac{\pi}{2} - \theta\right) = \cos(\theta) \), we simplify the computation of \( \sin(A) \) matrix. This transformation is crucial for accurately analyzing the properties of the resultant matrix. Without understanding these identities, it would be challenging to simplify expressions accurately.
Here are some key trigonometric identities you might find helpful:

• \( \sin^2(\theta) + \cos^2(\theta) = 1 \)
• \( \sin(2\theta) = 2\sin(\theta)\cos(\theta) \)
• \( \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) \)
• \( \sin(-\theta) = -\sin(\theta) \)

Understanding these can greatly aid in tasks such as proving whether a matrix is orthogonal, verifying identities, and checking invertibility of matrices derived from trigonometric operations.

Matrix Invertibility

Matrix invertibility is a significant property in linear algebra, determining whether a matrix can be reversed or undone. A matrix is said to be invertible if there exists another matrix which, when multiplied with the original matrix, results in an identity matrix. This is a vital concept because it underpins many operations in engineering, physics, and computer science operations involving linear transformations.
To determine if a matrix like \( \sin A \) is invertible, we calculate its determinant. In our problem, the determinant of \( \sin A \) is calculated using trigonometric identities:\[ \det(\sin A) = \cos^2(\theta) + \sin^2(\theta) = 1 \]The non-zero determinant confirms that the matrix is invertible.
Keep in mind these:

• A square matrix is invertible if and only if its determinant is non-zero.
• For a 2x2 matrix \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \), the matrix is invertible if \( ad - bc eq 0 \).

Recognizing when matrices can be inverted is crucial for tasks involving calculation solutions to systems of linear equations, invertibilty also allows for the calculation of matrix divisions used to solve vector transformations.

Orthogonal Matrices

An orthogonal matrix is a square matrix whose transpose equals its inverse. This property implies that when an orthogonal matrix is multiplied by its transpose, you obtain the identity matrix. Orthogonal matrices are critical in preserving distances and angles, making them valuable in computations requiring minimal numerical errors, such as in graphics and signal processing.
For a matrix like \( \sin A \), checking for orthogonality involves ensuring:\[ (\sin A)^T \sin A = I \]In our matrix calculations, it is confirmed by:\[ \begin{pmatrix} \cos(\theta) & \sin(\theta) \-\sin(\theta) & \cos(\theta) \end{pmatrix}\begin{pmatrix} \cos(\theta) & -\sin(\theta) \\sin(\theta) & \cos(\theta) \end{pmatrix} = I \]represents an orthogonal matrix.
Why are orthogonal matrices important?

• They preserve the lengths of vectors, which means they are ideal for geometric transformations, such as rotations and reflections.
• They have simple and efficient numerical properties, which is advantageous in minimizing rounding errors in computations.

Understanding orthogonality can be pivotal in domains where preserving the integrity of data through transformations is essential.