Problem 59 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The value of the determinant is \(\left|\begin{array}{ccc}\beta \gamma & \beta \gamma^{\prime}+\beta^{\prime} \gamma & \beta^{\prime} \gamma^{\prime} \\\ \gamma \alpha & \gamma \alpha^{\prime}+\gamma^{\prime} \alpha & \gamma^{\prime} \alpha^{\prime} \\ \alpha \beta & \alpha \beta^{\prime}+\alpha^{\prime} \beta & \alpha^{\prime} \beta^{\prime}\end{array}\right|\) (A) \(\left(\alpha \beta^{\prime}-\alpha^{\prime} \beta\right)\left(\beta \gamma^{\prime}-\beta^{\prime} \gamma\right)\left(\gamma \alpha^{\prime}-\gamma^{\prime} \alpha\right)\) (B) \(\alpha \beta \gamma(\alpha+\beta+\gamma)\left(\alpha^{\prime}+\beta^{\prime}+\gamma^{\prime}\right)\) (C) \(\alpha^{\prime} \beta^{\prime} \gamma^{\prime}(\alpha+\beta+\gamma)\left(\alpha^{\prime}+\beta^{\prime}+\gamma\right)\) (D) None of these

Step-by-Step Solution

Answer

(A) because the determinant simplifies to that expression using the factorization method.

Step 1: Identify the Matrix

The given determinant is a 3x3 matrix:\[ A = \begin{bmatrix} \beta \gamma & \beta \gamma^{\prime}+\beta^{\prime} \gamma & \beta^{\prime} \gamma^{\prime} \ \gamma \alpha & \gamma \alpha^{\prime}+\gamma^{\prime} \alpha & \gamma^{\prime} \alpha^{\prime} \ \alpha \beta & \alpha \beta^{\prime}+\alpha^{\prime} \beta & \alpha^{\prime} \beta^{\prime} \end{bmatrix} \]

Step 2: Concept of Determinant for a 3x3 Matrix

The determinant of a 3x3 matrix \( \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} \) is computed as:\[ |A| = a(ei - fh) - b(di - fg) + c(dh - eg) \]

Step 3: Compute the Determinant

Applying the determinant formula to our matrix:\[ \left|\begin{array}{ccc}\beta \gamma & \beta \gamma^{\prime}+\beta^{\prime} \gamma & \beta^{\prime} \gamma^{\prime} \ \gamma \alpha & \gamma \alpha^{\prime}+\gamma^{\prime} \alpha & \gamma^{\prime} \alpha^{\prime} \ \alpha \beta & \alpha \beta^{\prime}+\alpha^{\prime} \beta & \alpha^{\prime} \beta^{\prime}\end{array}\right| \]Calculate each term:- For the first term: \( \beta \gamma \left((\gamma \alpha^{\prime}+\gamma^{\prime} \alpha)(\alpha^{\prime} \beta^{\prime}) - (\gamma^{\prime} \alpha^{\prime})(\alpha \beta^{\prime}+\alpha^{\prime} \beta)\right) \)- For the second term: \( - (\beta \gamma^{\prime}+\beta^{\prime} \gamma) \left(\gamma \alpha \alpha^{\prime} \beta^{\prime} - \gamma^{\prime} \alpha^{\prime} \alpha \beta \right) \)- For the third term: \( \beta^{\prime} \gamma^{\prime} (\gamma \alpha(\alpha \beta^{\prime}+\alpha^{\prime} \beta) - \gamma \alpha^{\prime}\alpha \beta ) \)

Key Concepts

Matrix

Matrices are a powerful mathematical tool used to organize and manipulate numbers, expressions, or equations in a structured way. Each matrix consists of elements arranged in rows and columns.

In practical terms, matrices help in solving linear equations, performing transformations, and many other mathematical computations.

A matrix is denoted by square brackets or parentheses and is typically represented as

• ">A or M for matrices.


• A matrix with 'm' rows and 'n' columns is called an m x n matrix.

Matrices are widely used across various scientific fields including physics, engineering, computer graphics, and statistics. Understanding how they work is crucial for any advanced application.

3x3 Matrix

A 3x3 matrix is a common and important type of matrix in mathematics. It consists of three rows and three columns, making a total of nine elements.

You might see a generic 3x3 matrix represented as: \[ \begin{bmatrix} a & b & c \d & e & f \g & h & i \end{bmatrix} \]
In this arrangement, each element in the matrix is specifically located by row and column indices.

Understanding and operating with a 3x3 matrix is foundational for higher-level mathematics. It forms the basis for vector transformations, rotations in space, and more complex operations in linear algebra. 3x3 matrices are particularly significant due to their use in determining volumes and solving systems of equations.

Determinant Formula

The determinant of a matrix is a special value computed from its elements. It plays a pivotal role in linear algebra, particularly in equations involving transformations and systems of linear algebraic equations.

The determinant of a 3x3 matrix \( \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} \) is calculated using the formula:
\[|A| = a(ei - fh) - b(di - fg) + c(dh - eg) \]
To compute the determinant, follow these steps:

• Multiply the diagonal element combinations across the rows and columns.


• Apply the signs and sum up the products accounting for the structure (positive and negative signs as arranged).


• The result is the determinant which provides insights into properties like invertibility and volume scaling of a matrix.

The determinant tells us if a matrix is invertible, as a non-zero determinant indicates an invertible matrix. In these computations, it's crucial to work systematically to avoid errors and ensure accuracy.