For what values of λdoes the following set of equations have...

For what values of $\lambda$ does the following set of equations have nontrivial solutions for $x, y$. z? For each value of $\lambda$ find the corresponding solutions for $x, y, z$. (Comment: This is an example of what is called an eigenvalue or characteristic value problem in mathematica] physics; the values of $\lambda$ are the eigenvalues. See Chapters 10 and $12 .$ ) $$ \left\{\begin{array}{r} -(1+\lambda) x+y+3 z=0 \\ x+(2-\lambda) y=0 \\ 3 x+(2-\lambda) z=0 \end{array}\right. $$

For what values of $\lambda$ does the following set of equations have nontrivial solutions for $x, y$. z? For each value of $\lambda$ find the corresponding solutions for $x, y, z$. (Comment: This is an example of what is called an eigenvalue or characteristic value problem in mathematica] physics; the values of $\lambda$ are the eigenvalues. See Chapters 10 and $12 .$ )
$$
\left\{\begin{array}{r}
-(1+\lambda) x+y+3 z=0 \\
x+(2-\lambda) y=0 \\
3 x+(2-\lambda) z=0
\end{array}\right.
$$

Key Concepts

Characteristic Polynomial

The characteristic polynomial is derived by setting the determinant of the matrix (modified by subtracting lambda times the identity matrix) to zero. Solving this polynomial yields the eigenvalues of the matrix, which are the specific values of the parameter for which the system admits nontrivial solutions.

Eigenvector Determination

Once the eigenvalues are obtained, the corresponding eigenvectors are found by substituting each eigenvalue back into the original system of equations and solving for the variables. The eigenvectors represent directions in the vector space where the transformation only scales without changing the direction.

Eigenvalue Problem

An eigenvalue problem is a central concept in linear algebra and functional analysis where one looks for numbers (eigenvalues) and nonzero vectors (eigenvectors) such that when a linear transformation (or matrix) is applied to the vector, the result is the scalar multiple of that vector. This problem is fundamental in many areas, including physics and engineering, for understanding system dynamics.

Determinant and Nontrivial Solutions

In the context of homogeneous systems of linear equations, a nontrivial solution exists if and only if the determinant of the coefficient matrix is zero. This condition ensures that the system has an infinite number of solutions and is indicative of linear dependence among the equations.

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