If the system of equations a x+b y+c=0, b x+c y+a =0, c x+a...

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

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

Key Concepts

Determinants

The determinant of a square matrix is a scalar value that provides important information about the matrix, including whether it is invertible (nonzero determinant) or singular (zero determinant). In the context of linear systems, a nonzero determinant indicates a unique solution, while a zero determinant suggests that the system may have infinitely many solutions or be inconsistent, depending on additional constraints.

Matrix Rank

The rank of a matrix is the maximum number of linearly independent rows or columns in the matrix. It is directly related to the number of free variables present in the system of equations. In solving linear systems, knowing the rank helps determine whether the system has a unique solution, infinitely many solutions, or no solution at all, by comparing it with the number of variables.

Systems of Linear Equations

This concept involves multiple linear equations that contain several variables. The goal is to find values of the variables that satisfy all the equations simultaneously. Methods for solving these systems include substitution, elimination, and matrix-based methods, each of which relies on the inherent structure of linear relations.

Homogeneous Systems

A homogeneous system of linear equations is one in which all constant terms are zero. Such systems always have the trivial solution (all variables equal to zero), and they may have additional, nontrivial solutions if the system’s coefficient matrix is singular. This property is crucial in determining whether the system has a unique solution or infinitely many solutions.

Consistency Conditions of Linear Systems

A system of linear equations is consistent if there exists at least one solution. Analyzing the relationships among the equations, such as via the coefficient matrix and augmented matrix, allows one to determine if the system meets the conditions for consistency. In homogeneous systems, consistency is automatically satisfied, while in nonhomogeneous systems, restrictions on parameters may need to be met.

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