Problem 39 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If \(a, b, c\) are sides of a scalene triangle, then the value of \(\left|\begin{array}{lll}a & b & c \\ b & c & a \\ c & a & b\end{array}\right|\) is : [OnlineApril 9, 2013] (a) non-negative (b) negative (c) positive (d) non-positive

Step-by-Step Solution

Answer

The determinant is negative.

Step 1: Title

Consider the given determinant of a 3x3 matrix where the sides of a scalene triangle are the elements.

Step 2: Calculate the Cofactor Expansion

Perform cofactor expansion (Laplace expansion) on the matrix along the first row: \[|a(b^2-c^2) - b(b^2-c^2) + c(b^2-c^2)|\] That implies simplifying the determinant using terms of second and third rows.

Step 3: Simplify the Expression

Recognize and simplify \[ a(b(c-a) - b(b-c)) - b(c(a-b) - b(c-a)) + c((b^2-ca)-(bc-ab))\] which further reduces using identity properties of symmetric functions.

Step 4: Evaluate the Determinant

By reducing the determinant with the properties of alternants and plugging in the identity terms \[= (a+b+c)(-2abc)\] this tells us the final result retains negative sign due to the alternating symmetric functions.

Key Concepts

Matrix Cofactor Expansion

Cofactor expansion, also known as Laplace expansion, is a method used to calculate the determinant of a matrix, especially helpful when dealing with matrices larger than 2x2. The essential idea is to break down a complex determinant into smaller, more manageable calculations.

Here's how it works:

• You choose a specific row or column from the matrix.
• For each element in that row or column, compute the minor — the determinant of the submatrix that remains after removing the row and column of the element.
• Multiply each element by its corresponding minor and the sign factor. The sign is determined by the position of the element using the sign rule "+" if the sum of row and column indices (i+j) is even, "-" if odd.
• Sum all these values together to get the determinant of the matrix.

This method simplifies computations by breaking the problem into smaller parts. For our problem, the cofactor expansion along the first row is essential in developing the linear combinations used to simplify the determinant of the symmetrical matrix involving triangle sides.

Properties of Determinants

Determinants have several properties that make them incredibly useful for solving linear algebra problems:

• The determinant of a matrix changes sign if any two rows or columns are swapped.
• The determinant of a matrix is zero if two rows or columns are identical.
• Scaler multiplication of any row or column scales the determinant by the same factor.
• Determinants can be directly computed by Laplace expansion in terms of any row or column, not just the first one.
• Importantly, for similar matrices (matrices that are transformations of one another), the determinants will be equal.

These properties, when used together, can help simplify complex matrices, as in our example, where the symmetry of the matrix related to triangle sides is paired with properties such as linearity and alternants identity. This allows us to simplify the determinant drastically to ascertain the sign.

Symmetric Functions in Algebra

Symmetric functions play a crucial role in understanding matrix behavior, especially when calculating determinants of matrices that have a symmetrical structure. These functions are invariant under permutations of their variables.

For example, consider a 3-variable polynomial function where the variables are the sides of a triangle. Symmetric functions can be classified as:

• Sum of variables: e.g., \(a + b + c\)
• Product of all variables: e.g., \(abc\)
• Sum of product pairs: e.g., \(ab + bc + ca\)

Understanding these forms helps in simplifying polynomials. In the determinant from our exercise, symmetric alternant exploitation, such as recognizing multiple products and sums, leads us to deduce that the determinant's expression simplifies to a straightforward form, involving the product of perimeter \(a + b + c\) and symmetric alternant \(-2abc\). This understanding provides insight into why our solution has a particular sign.