Problem 84 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(A+B+C=\pi, e^{i \theta}=\cos \theta+i \sin \theta\) and \(z=\left|\begin{array}{lll}e^{2 i A} & e^{-i C} & e^{-i B} \\ e^{-i C} & e^{2 i B} & e^{-i A} \\ e^{-i B} & e^{-i A} & e^{2 i C}\end{array}\right|\) then (A) \(\operatorname{Re}(z)=4\) (B) \(\operatorname{Im}(z)=0\) (C) \(\operatorname{Re}(z)=-4\) (D) \(\operatorname{Im}(z)=-1\)

Step-by-Step Solution

Answer

The correct option is (A) \( \operatorname{Re}(z)=4 \).

Step 1: Understand Given Information

We are given a matrix \( z \) and the conditions that \( A + B + C = \pi \) and that the exponential form \( e^{i \theta} = \cos \theta + i \sin \theta \). The problem involves finding the determinant of a matrix with complex exponentials.

Step 2: Simplify Using Trigonometric Identities

Since \( A + B + C = \pi \), we can use this to simplify the components of the matrix using the identities of complex numbers and the unit circle properties. For angles summing to \( \pi \), trigonometric identities can provide simplifications.

Step 3: Calculate the Determinant

The determinant of a 3x3 matrix is given by:\[| M | = a(ei - fh) - b(di - fg) + c(dh - eg)\]where \( M \) is the 3x3 matrix, and \( a, b, c, ..., i \) are the elements of the matrix. Apply this formula to our matrix \( z \) replacing each exponential term with its corresponding identity.

Step 4: Use Symmetrical Properties

Given the symmetrical structure of the matrix relative to its main diagonal and the identity \( A + B + C = \pi \), main diagonals will negate the exponentials combined with off-diagonal terms due to symmetry. Therefore every exponential term contributes to reversing its sign, simplifying to constant real values.

Step 5: Conclusion on Real Part

The simplification using symmetry and properties of the determinant shows the calculation yields \( \Re(z) = 4 \). Since off-diagonal terms cancel out, we know \( \Im(z) = 0 \) and the real part dominates.

Key Concepts

Trigonometric Identities

When working with angles, trigonometric identities simplify complex problems, especially in exercises related to complex numbers and matrices. A core identity is the Pythagorean identity:

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

This identity helps in dealing with expressions involving sine and cosine, such as those we encounter in our problem, where \( A + B + C = \pi \). It means that every angle in the identity adds up to half a full circle. This identity can be extended with other identities like:

• \( \sin(A + B) = \sin A \cos B + \cos A \sin B \)
• \( \cos(A + B) = \cos A \cos B - \sin A \sin B \)

These formulas help simplify the complex exponential terms in matrix entries by expressing them in terms of sine and cosine. Breaking down exponential forms such as \( e^{i\theta} \) into its trigonometric components is crucial for solving problems that involve complex numbers and helps understand their geometric representation on the unit circle.

Matrix Determinant

To solve problems involving matrices with complex numbers, understanding how to compute a matrix determinant is essential. For a 3x3 matrix, the formula for determining its determinant \( |M| \) is:\[ |M| = a(ei - fh) - b(di - fg) + c(dh - eg) \]where \( a, b, c, ..., i \) represent the elements of the matrix. Here, this formula calculates the value that will characterize the given matrix. Computation of the determinant is important: it can reveal many properties such as invertibility and volume scaling of linear transformations.
In our context, this determinant involves complex exponentials like \( e^{2iA} \), but by simplifying using trigonometric identities, we can express them as sums or differences of simpler trigonometric functions. Notably, symmetrical matrices, where rows and columns mirror one another, can simplify determinant calculations considerably, as seen in our example problem.

Euler's Formula

Euler's Formula is a beautiful equation connecting complex numbers and trigonometry, expressed as:

• \( e^{i\theta} = \cos \theta + i\sin \theta \)

It elegantly bridges the gap between exponential forms and trigonometric functions. In our problem, this formula helps convert the exponential elements of our matrix to something more tangible involving real numbers. This simplification is crucial because it allows us to harness properties and identities of sine and cosine functions to solve complex matrix problems.
Using Euler's Formula, the elements such as \( e^{2iA} \) can be interpreted geometrically as rotations on the complex plane, which translates angles into exponential terms. Thus, it provides a comprehensive tool to deal with both real and imaginary components effectively, simplifying the calculations significantly in problems like ours.

Complex Exponentials

Complex exponentials are expressions of the form \( e^{i\theta} \), which encapsulate both real and imaginary aspects, neatly modeled by Euler's Formula. These exponentials turn angles into complex numbers on the unit circle, allowing us to utilize trigonometric identities for simplification.
In matrix problems, each member of a matrix like \( z \) can be represented using complex exponentials, simplifying multiplicative processes as these can be split into sums and differences of sines and cosines. These transformations ease our computation of determinants. The symmetry observed in matrix formulations means exploiting the cancellation of terms involving \( e^{-iB} \) and similar expressions, which simplifies our expression down to a real constant, explaining why the imaginary component \( \Im(z) \) would nullify in our specific exercise.
This use of complex exponentials is integral in balancing and converting the complex numbers inherent in matrix problems into something that reveals clear, real numbers, which is crucial in reaching the solution accurately.