Let's go through each question one at a time and analyze the options provided.

Question 1:

The system of equations given by: $2x_1 + 2x_2 + x_3 = 5,$ $x_1 + 3x_2 + x_3 = 2,$ $3x_1 + 4x_2 + 5x_3 = 1$ is to be written in matrix form as:

This system of equations can be expressed in matrix form as: $\begin{bmatrix} 2 & 2 & 1 \\ 1 & 3 & 1 \\ 3 & 4 & 5 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} 5 \\ 2 \\ 1 \end{bmatrix}$

Looking at the answer choices, the correct option is: $\begin{bmatrix} 2 & 2 & 1 \\ 1 & 3 & 1 \\ 3 & 4 & 5 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} 5 \\ 2 \\ 1 \end{bmatrix}$

Question 2:

The augmented matrix for the system of equations: $2x_1 + 2x_2 + x_3 = 5,$ $x_1 + 3x_2 + x_3 = 2,$ $3x_1 + 4x_2 + 5x_3 = 1$ is requested.

The augmented matrix combines the coefficients and constants from the equations, resulting in: $\begin{bmatrix} 2 & 2 & 1 & | & 5 \\ 1 & 3 & 1 & | & 2 \\ 3 & 4 & 5 & | & 1 \end{bmatrix}$

The correct answer matches this augmented matrix structure.

Would you like further clarification or a detailed explanation of any part of this solution?

Here are some additional related questions to explore:

1. How can we solve a system of equations using matrix inversion?
2. What are the properties of an augmented matrix in a system of linear equations?
3. How does Gaussian elimination apply to solving systems with augmented matrices?
4. What does it mean if the determinant of the coefficient matrix is zero?
5. How can we represent an inconsistent system in augmented matrix form?

Tip: Understanding matrix representation of systems of equations simplifies solving large systems by using row reduction or matrix operations.