Problem 66 - A Complete Resource Book in Mathematics for JEE Main

Question

The value of the determinant \(\left|\begin{array}{ccc}(b+c)^{2} & c^{2} & b^{2} \\ c^{2} & (c+a)^{2} & a^{2} \\ b^{2} & a^{2} & (a+b)^{2}\end{array}\right|\) is (A) \(2(a b+b c+c a)^{3}\) (B) \((a b+b c+c a)^{3}\) (C) \(4(a b+b c+c a)^{3}\) (D) None of these

Step-by-Step Solution

Answer

The answer is \(2(a b + b c + c a)^3\) which corresponds to option (A).

Step 1: Recognize the Problem Type

This is a problem involving the calculation of the determinant of a 3x3 matrix. The elements of the matrix are expressions in terms of variables \(a\), \(b\), and \(c\).

Step 2: Simplify the Matrix

Notice that the structure of the matrix suggests symmetry and dependent relation between rows and columns based on the squared terms. This suggests that reducing or simplifying the matrix or deterministic strategy (such as considering specific values) might help.

Step 3: Calculate Row Operations

Perform row operations on the matrix to simplify it. Subtract the second column from the first and the third column from the second, effectively transforming the matrix into one that is easier to compute.

Step 4: Evaluate the Simplified Determinant

After the row operations, check the results and evaluate the determinant of the reduced form. If correctly simplified, the resulting structure should aid in directly calculating the determinant.

Step 5: Compare and Confirm Results

Compare your calculations from the simplified structure and cross-check with the options provided. You should ensure the determinant matches one of the given forms.

Step 6: Select the Correct Option

After evaluating the determinant and simplifying, the answer should match one of the options. The correct expression of this determinant is \(2(a b + b c + c a)^3\).

Key Concepts

Matrix Operations

When dealing with matrices, operations like addition, subtraction, and multiplication often come into play. However, calculating determinants introduces a unique set of matrix operations focused on reducing complexity. Determinants can help verify matrix solvability and understanding them can be crucial for solving linear equations.

Performing matrix operations often involves strategic maneuvers like row operations. These include:

• Swapping rows
• Adding a multiple of one row to another
• Multiplying a row by a constant factor

Applying these operations correctly can transform a complex matrix into a simpler one, making determinant calculations more accessible. This is particularly significant in a 3x3 matrix where every simplification can vastly reduce computational effort.

3x3 Matrix

A 3x3 matrix is a square matrix with three rows and three columns. It is one of the most manageable matrix sizes for hand calculations while still containing enough complexity to explore interesting mathematical properties. The determinant of a 3x3 matrix is calculated using a specific formula that reflects the matrix's internal structure. For a matrix given as:\[\begin{bmatrix}a & b & c \d & e & f \g & h & i\end{bmatrix}\]the determinant is calculated as:\[a(ei - fh) - b(di - fg) + c(dh - eg)\]This formula results from expanding the matrix across a row or a column, each entry being multiplied by its minor and adjusted by a factor of "+" or "-" depending on its position.

Understanding this concept is key when working with matrices, as it lays the foundation for more advanced topics such as eigenvalues, eigenvectors, and diagonalization.

Expression Simplification

Expression simplification is a valuable technique in mathematics which can transform complex expressions into more manageable forms. This process is essential when calculating determinants, especially for matrices containing algebraic expressions. By simplifying each expression, you decrease the potential for error and make subsequent calculations easier.

A vital part of simplification is recognizing patterns or symmetry within the expressions. Observing how terms relate or repeat can guide you to a more straightforward path to the solution. In our original exercise, by initially simplifying the matrix using row operations, the determinant calculation is simplified. This step is vital as it helps identify equivalent expressions or values that can reduce redundant calculations.

Embracing simplification can reveal solutions more clearly and ensure mathematical operations are both efficient and effective.