It looks like you have a set of mathematical problems related to linear algebra. Let's break them down:

Q1: For what values of λ do the following homogeneous equations have non-trivial solutions?

We are given the following system of equations:

$(3 - \lambda)x_1 - x_2 + x_3 = 0$ $x_1 + ( - \lambda + 2)x_2 + x_3 = 0$ $x_1 - x_2 + (1 - \lambda)x_3 = 0$

This is a homogeneous system, and the solutions exist if the determinant of the coefficient matrix is 0. You will need to find the determinant of the corresponding matrix and solve for λ.

Q2: In the following, use the Gauss-Jordan method to reduce the given system to reduced echelon form, indicating the operations performed and determining the solution.

The system is:

$5x_1 + 5x_2 - x_3 = 0$ $10x_1 + 5x_2 + 2x_3 = 0$ $5x_1 + 15x_2 - 9x_3 = 0$

You will apply the Gauss-Jordan elimination method to reduce this system to row echelon form and then solve for the variables.

Q3: Find the eigenvalues of the given matrices.

You are asked to find the eigenvalues of these two matrices: