Determine all values of the constant k for which the...

Name: Problem 24, Chapter 2: Differential Equations and Linear Algebra
Uploaded: 2020-06-22T20:39:36-08:00
Duration: 6 min 52 s
Channel: Jingyun Wang
Description: Determine all values of the constant k for which the following system has (a) no solution, (b) an infinite number of solutions, and (c) a unique solution. x1+2 x2-x3 =3 2 x1+5 x2+x3 =7 x1+x2-k^2 x3 =-k

Determine all values of the constant $k$ for which the following system has (a) no solution, (b) an infinite number of solutions, and (c) a unique solution.
$$
\begin{aligned}
x_{1}+2 x_{2}-x_{3} &=3 \\
2 x_{1}+5 x_{2}+x_{3} &=7 \\
x_{1}+x_{2}-k^{2} x_{3} &=-k
\end{aligned}
$$

Key Concepts

Parameter Analysis in Linear Systems

When a system of linear equations contains a parameter, the value of this parameter can alter the properties of the system, such as the rank of the matrices involved. Analyzing how the parameter affects the system is crucial to determining the cases for unique solutions, infinite solutions, or no solutions, as different parameter values may lead to different types of degeneracies in the system.

Systems of Linear Equations

A system of linear equations is a collection of equations involving the same set of variables. The solutions to such systems represent the set of variable assignments that satisfy all equations simultaneously. The analysis of these systems often involves determining whether there is a unique solution, infinitely many solutions, or no solution at all.

Matrix Representation and Augmented Matrices

Systems of equations can be represented in matrix form, where the coefficients of the variables form the coefficient matrix and the constants form the augmented column. This representation facilitates the use of systematic procedures such as Gaussian elimination to study the structure of the solutions.

Determinants and Invertibility

The determinant of a square matrix is a scalar value that indicates whether the matrix is invertible. A non-zero determinant implies that the matrix is invertible and the corresponding system has a unique solution. Conversely, a zero determinant suggests that the matrix is singular, which may lead to either infinitely many solutions or no solution, depending on other consistency conditions.

Rouché-Capelli Theorem (Consistency Conditions)

The Rouché-Capelli theorem provides a method for determining the consistency of a system of linear equations. It states that a system is consistent if and only if the rank of the coefficient matrix equals the rank of the augmented matrix. This theorem is fundamental in deciding when a system will have a unique solution, infinitely many solutions, or no solution.

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