Problem 87 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(\left|\begin{array}{ccc}b c-a^{2} & c a-b^{2} & a b-c^{2} \\ c a-b^{2} & a b-c^{2} & b c-a^{2} \\ a b-c^{2} & b c-a^{2} & c a-b^{2}\end{array}\right|=\left|\begin{array}{ccc}\alpha^{2} & \beta^{2} & \beta^{2} \\ \beta^{2} & \alpha^{2} & \beta^{2} \\ \beta^{2} & \beta^{2} & \alpha^{2}\end{array}\right|\) then (A) \(\alpha^{2}=a^{2}+b^{2}+c^{2}\) (B) \(\beta^{2}=a b+b c+c a\) (C) \(\alpha^{2}=a b+b c+c a\) (D) \(\beta^{2}=a^{2}+b^{2}+c^{2}\)

Step-by-Step Solution

Answer

The correct match is option (A): \(\alpha^{2}=a^{2}+b^{2}+c^{2}\).

Step 1: Identify Matrix Type

Both matrices are symmetric, implying their determinant can be derived using specific patterns. Recognize that the matrix on the right is of the form \(A - B\), where \(A\) and \(B\) have structured entries.

Step 2: Compute Determinant of Right Matrix

The right matrix is \( \begin{bmatrix} \alpha^2 & \beta^2 & \beta^2 \ \beta^2 & \alpha^2 & \beta^2 \ \beta^2 & \beta^2 & \alpha^2 \end{bmatrix} \), which can be rewritten by factoring. Its determinant is given by the formula for symmetric matrices: \(2\alpha^2\beta^2 + \beta^6 - \alpha^4 - 2\alpha^2\beta^2 = (\alpha^2 + 2\beta^2)(\alpha^2 - \beta^2)^2\).

Step 3: Compute Determinant of Left Matrix

Let \(x = bc - a^2\), \(y = ca - b^2\), and \(z = ab - c^2\) for simplicity. The left matrix \( \begin{bmatrix} x & y & z \ y & z & x \ z & x & y \end{bmatrix} \) also uses the same determinant formula pattern: \(2xyz + x^3 + y^3 + z^3 - 3xyz\).

Step 4: Equate Determinants

Set the determinants of both matrices equal: \((\alpha^2 + 2\beta^2)(\alpha^2 - \beta^2)^2 = 2xyz + x^3 + y^3 + z^3 - 3xyz\). Determine \(\alpha^2\) and \(\beta^2\) so that this equality holds under the assumption of the structure.

Step 5: Test Options Based on Derived Factors

Consider the options given (A, B, C, D) and check them against the results of expanding and comparing the two determinant calculations. By aligning the forms, match the simplified outcomes of \(\alpha^2\) and \(\beta^2\) with the provided choices.

Key Concepts

Determinants

Determinants are a fundamental concept in linear algebra, particularly when solving matrix problems. A determinant provides a scalar value that is particularly useful in systems of linear equations, as well as understanding certain properties of matrices like invertibility.
To compute the determinant of an n x n matrix, one can use a recursive method that breaks down the matrix into smaller matrices. This is typically done using an expansion by minors and cofactors. For 3x3 matrices, like those in the exercise, there is a specific formula involving the sum and difference of products of the elements of the matrix.

• The determinant can help identify if a matrix is invertible - a non-zero determinant indicates invertibility.
• Determinants exhibit specific behaviors under operations such as swaps of rows or columns, scaling of rows or columns, and adding multiples of rows to others.

Understanding determinants is crucial in the solutions of matrix equations, and here, plays a key role in equating two matrix expressions.

Symmetric Matrices

Symmetric matrices are square matrices that are identical to their transposes, which means that they look the same on both sides of the main diagonal. This special property simplifies many computational problems, as symmetric matrices possess unique characteristics.
In the exercise, both matrices involved are symmetric. This symmetry means that certain simplifications are possible when computing their determinants, as symmetric matrices tend to have patterns that repeat due to their mirrored entries.

• Symmetric matrices have real eigenvalues, making them important in various applications.
• The symmetry simplifies computations, as the elements above the main diagonal are equal to those below it.

Recognizing symmetry in a matrix can be an advantage during computations like determinant calculation and solving matrix equations.

Matrix Equations

Matrix equations are composed of matrices and are often used to represent systems of linear equations. Solving matrix equations involves finding matrices or scalars that satisfy the equation.
In the given exercise, the challenge is to find values for \(\alpha^2\) and \(\beta^2\) that satisfy the determinant equation.

• One common method for solving matrix equations is to equate determinants when both sides of the equation involve matrices that have the same form or structure.
• Matrix equations can often be manipulated algebraically, similar to standard algebraic equations, to isolate variables or simplify expressions.

Understanding how matrix equations function and how to manipulate them accordingly is essential in fields like engineering and computer science.

Alpha and Beta Values

In the exercise, \(\alpha\) and \(\beta\) are variables that need to be determined based on the equivalence of two matrix determinants. The values of \(\alpha^2\) and \(\beta^2\) have options provided, and understanding how to verify or derive these values requires knowledge of the matrices involved.
The technique involves setting known expressions and expanded terms of the determinant equal to each other and solving for the variables.

• By substituting various assumptions about the relations of matrix elements, one can test which option leads to consistency in the determinant equation.
• Such problems often require testing under options provided or additional mathematical properties like matrix ranks, eigenvalues, etc.

Mastery of these techniques helps in many branches of mathematical problem-solving, from purely theoretical work to applied mathematics.