Problem 26 - A Complete Resource Book in Mathematics for JEE Main

Question

If the system of equations \(a x+b y+c=0, b x+c y+a\) \(=0, c x+a y+b=0\) has a solution then the system of equations \((b+c) x+(c+a) y+(a+b) z=0\) \((c+a) x+(a+b) y+(b+c) z=0\) \((a+b) x+(b+c) y+(c+a) z=0\) has (A) only one solution (B) no solution (C) infinite number of solutions (D) None of these

Step-by-Step Solution

Answer

The second system has an infinite number of solutions (Option C).

Step 1: Identify Given Equation System

We are initially given the system of equations: \( ax + by + c = 0 \), \( bx + cy + a = 0 \), and \( cx + ay + b = 0 \). We have to ensure these equations have a solution.

Step 2: Formulate Homogeneous System

Rewrite the first system into a homogeneous form by letting \( z = 1 \). The equations become: \( ax + by + cz = 0 \), \( bx + cy + az = 0 \), and \( cx + ay + bz = 0 \). This implies these equations are linearly dependent, meaning there is a non-trivial solution.

Step 3: Analyze Second System of Equations

Consider the second system: \((b+c) x + (c+a) y + (a+b) z = 0\), \((c+a) x + (a+b) y + (b+c) z = 0\), \((a+b) x + (b+c) y + (c+a) z = 0\). Assess the possibility of solutions.

Step 4: Determine Coefficient Matrix and Rank

Construct the coefficient matrix for the second system and calculate its determinant or rank. Recognize that the system coefficients form a symmetric pattern. Such systems are known to be homogenous with the property that the sum of coefficients for each equation results in zero, giving us a chance for infinite solutions.

Step 5: Conclude on Solution Existence

Since both systems have the same structure in terms of symmetry and linear dependence, they allow for an infinite number of solutions given that the sum of coefficients in each equation of the second system is zero.

Key Concepts

Linear Dependence

Linear dependence is a fundamental concept in linear algebra. In the context of systems of linear equations, it refers to a set of equations where one equation can be expressed as a combination of the others. If this situation arises, the system of equations is termed as having linearly dependent equations.

When applied to the exercise, the first given system of equations, upon transformation, reveals itself to be linearly dependent. This is evident since each equation is not unique and can be derived from the others in the set. Identifying linear dependence is essential because it often hints at the existence of either no solutions or infinitely many solutions for the system. In the example above, recognizing this linear dependence helps us assert that there are non-trivial solutions, that is, solutions beyond the trivial solution of all variables being zero.

Homogeneous Systems

A homogeneous system of linear equations refers to a system where all the constant terms are zero. Such systems generally take the form:

• First equation: \( ax + by + cz = 0 \)
• Second equation: \( bx + cy + az = 0 \)
• Third equation: \( cx + ay + bz = 0 \)

The main attribute of homogeneous systems is their potential to have infinite solutions, particularly when the rank of the coefficient matrix is less than the number of variables. In this problem, the given first system, originally non-homogeneous, is turned into a homogeneous one to analyze its solutions.

When transformed back into its homogeneous equivalent by adding a new variable \( z \) as 1, the system's behavior is analyzed. The transformation allows us to explore whether solutions are possible and, if so, the nature of these solutions based on the coefficients' relationships.

Symmetric Matrices

Symmetric matrices are another crucial concept in linear algebra, playing a significant role in systems of equations. A matrix is considered symmetric if it is identical to its transpose, meaning the elements mirror along the main diagonal. In this exercise, the coefficients of the second system exhibit a symmetric pattern.

A significant property of symmetric matrices in the context of linear equations is that they tend to suggest stability and predictable outcomes with regards to solutions. Because of this symmetry, sometimes additional properties emerge, such as easy determination of eigenvalues or showing particular patterns like rows that sum to zero, which greatly affects the system's solutions availability.
Recognizing this symmetry helps infer the nature of solutions - either no solution or infinite solutions, as in the case we're examining.

Infinite Solutions

When dealing with linear systems of equations, one interesting outcome is obtaining infinite solutions. This typically occurs when the rank of the system's coefficient matrix is less than the number of variables. Remarkably, homogeneous systems have a non-trivial solution if this condition is met.

• The equations align in such a manner that multiple solutions satisfy all equations simultaneously.
• In our given exercise, the structural symmetry of the second system aligns such that the row-wise sum to zero—an indicator of infinite solutions.

This insight is pivotal in determining the kind of solutions a system possesses. Therefore, infinite solutions indicate many solutions exist, which are often related by a scalar multiple in the solution set of vector form. Thus, understanding the environment giving rise to infinite solutions enhances grasping the vast range of possibilities available within such linear systems.