For which values of b and c does the system x1+x2+b x3=1, b x1+3 x2-x3=-2, 3 x1+4 x2+x3=c, have

step 1 Let's solve a linear system given by x1 plus x2 plus x3 equals A, 2x1 plus 2x3 equals B, and 3x3

For which values of b and c does the system x1+x2+b x3=1, b...

Key Concepts

System of Linear Equations

A system of linear equations consists of multiple linear equations involving the same set of variables. In the context of this problem, the system is expressed with parameters that affect the coefficients and constant terms, illustrating how variations in these parameters can alter the solution set of the equations.

Consistency and Inconsistency in Linear Systems

The notions of consistency and inconsistency refer to whether a system has at least one solution or none at all. A consistent system has at least one solution, either unique or infinitely many, while an inconsistent system has no solution. Determining consistency often involves analyzing the relationships between the equations, especially when parameters are present.

Unique and Infinitely Many Solutions

A system of equations has exactly one solution when the equations are independent, typically confirmed by a nonzero determinant or full rank of the coefficient matrix. Conversely, infinitely many solutions occur when there is a dependency among the equations, meaning the coefficient matrix does not have full rank and there exists a free variable, resulting in a parametric family of solutions.

Matrix Rank and the Rouché–Capelli Theorem

The rank of a matrix is the number of linearly independent rows or columns. The Rouché–Capelli theorem states that a system of linear equations is consistent if and only if the rank of the coefficient matrix equals the rank of the augmented matrix. This concept is crucial for determining whether the system has no solution, a unique solution, or infinitely many solutions based on the parameters influencing the matrix.

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