Problem 73 - A Complete Resource Book in Mathematics for JEE Main

Question

The value of the determinant \(\left|\begin{array}{ccc}\sin \alpha \cos \beta & \cos \alpha \cos \beta & -\sin \alpha \sin \beta \\ \sin \alpha \sin \beta & \cos \alpha \sin \beta & \sin \alpha \cos \beta \\ \cos \alpha & -\sin \alpha & 0\end{array}\right|\) is (A) is independent of \(\alpha\) (B) independent of \(\beta\) (C) independent of \(\alpha\) and \(\beta\) (D) None of these

Step-by-Step Solution

Answer

The determinant is independent of \( \alpha \) and \( \beta \), so the answer is (C).

Step 1: Interpret the Determinant Expression

Our goal is to determine if the determinant of the given matrix depends on the variables \( \alpha \) and \( \beta \). The matrix is a 3x3 matrix with elements that are trigonometric functions of \( \alpha \) and \( \beta \).

Step 2: Compute the Determinant

The determinant of a 3x3 matrix \( \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} \) is given by \( a(ei - fh) - b(di - fg) + c(dh - eg) \). Apply this formula to our matrix.

Step 3: Identify Elements of Matrix

The elements of the given matrix are: - \( a = \sin \alpha \cos \beta \)- \( b = \cos \alpha \cos \beta \)- \( c = -\sin \alpha \sin \beta \)- \( d = \sin \alpha \sin \beta \)- \( e = \cos \alpha \sin \beta \)- \( f = \sin \alpha \cos \beta \)- \( g = \cos \alpha \)- \( h = -\sin \alpha \)- \( i = 0 \)

Step 4: Substitute and Simplify

Substitute these into the determinant formula: \[ \sin \alpha \cos \beta (\cos \alpha \sin \beta \cdot 0 + \sin \alpha \cos \beta \cdot \sin \alpha) - \cos \alpha \cos \beta (\sin \alpha \sin \beta \cdot 0 + \sin \alpha \cos \beta \cdot \cos \alpha) + (-\sin \alpha \sin \beta)(\sin \alpha \sin \beta \cdot -\sin \alpha - \cos \alpha \sin \beta \cdot \cos \alpha) \]Simplify this expression to calculate the determinant value.

Step 5: Calculate Step by Step

Evaluate and simplify each component:1. The first term: \( \sin^2 \alpha \cos^2 \beta \). 2. The second term: \( -\cos^2 \alpha \cos^2 \beta \).3. The third term expanded: \( \sin \alpha \sin \beta(\sin^2 \alpha \cos \alpha + \cos^2 \alpha \sin \beta) \). Simplify: the second term cancels the first when combined, cancelling dependence on \( \beta \). In the third term, all \( \beta \) terms cancel.

Step 6: Analyze the Result

After simplification, the result is independent of \( \beta \), and since the \( \sin \alpha \) and \( \cos \alpha \) terms also cancel each other, leaving the determinant independent of both \( \alpha \) and \( \beta \). Thus, the determinant value is independent of both variables.

Key Concepts

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are fundamental in mathematics. They relate the angles of a triangle to the ratios of its sides in a right-angled triangle. In this exercise, trigonometric functions like \( \sin \alpha \) and \( \cos \beta \) are utilized to form the elements of the given 3x3 matrix.
These functions have specific periodic properties:

• \( \sin \theta \) represents the vertical y-coordinate of a point on the unit circle.
• \( \cos \theta \) represents the horizontal x-coordinate.
• They both take values between -1 and 1.

In this context, the trigonometric functions evaluate the elements based on angles \( \alpha \) and \( \beta \), influencing how we treat these elements when calculating the determinant. Interestingly, despite this initial dependence on \( \alpha \) and \( \beta \), after matrix simplification, the trigonometric terms cancel out, showcasing the power of these functions in transforming matrix calculations.

3x3 Matrix

A 3x3 matrix is a square array with three rows and three columns. Each entry in the matrix can be part of complex operations like finding determinants or solving systems of equations.
In this exercise, the matrix is populated with trigonometric function values, appearing as:

• The first row: \( \sin \alpha \cos \beta, \cos \alpha \cos \beta, -\sin \alpha \sin \beta \)
• The second row: \( \sin \alpha \sin \beta, \cos \alpha \sin \beta, \sin \alpha \cos \beta \)
• The third row: \( \cos \alpha, -\sin \alpha, 0 \)

For a 3x3 matrix, the determinant testifies about the volume or scaling factor the matrix imparts to the vectors. Importantly, solving this determinant helps in verifying if the matrix remains dependent on specific variables.
Upon simplification using the determinant formula, it was found that the effects of \( \alpha \) and \( \beta \) vanish, concluding that the matrix's determinant is independent of these angles.

Matrix Simplification

Matrix simplification is a critical process to reduce complexity when working with matrices. It involves using algebraic manipulations and properties like row operations or identity matrices.
In the given problem, matrix simplification starts by applying the formula for determining the determinant of a 3x3 matrix:
For a matrix \( \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} \), the determinant is calculated as: \[ a(ei - fh) - b(di - fg) + c(dh - eg) \]
Using this formula on our matrix, trigonometric terms began to interact and cancel out irrelevant parts as derivations progressed. Specifically, steps included grouping similar trigonometric terms and recognizing identities such as \( \sin^2 \theta + \cos^2 \theta = 1 \) which helped simplify the determinant to become independent of \( \alpha \) and \( \beta \).
This simplification highlights how intricately linked trigonometric identities and determinant rules are in producing a neat solution, often much simpler than the original expression suggests.