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The image contains a set of matrix and system of equation problems related to linear algebra. Here's a breakdown of the tasks presented:

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**A.** The first task involves determining the solution set for a homogeneous system of equations represented by the matrix:

\[
\begin{bmatrix}
1 & 5 & -2 & 0 & 3 \\
0 & 0 & 0 & 6 & 0 \\
0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0
\end{bmatrix}
\]

The question asks for the solution set under this form of row-reduced echelon matrix.

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**B.** The second task requires solving two systems of equations using Gaussian elimination to row-reduce the system:

1. The first system:
\[
\begin{aligned}
x_1 + x_2 + 3x_3 + x_4 &= 0 \\
x_1 - 2x_2 + 3x_3 + x_4 &= 0 \\
x_1 - 5x_2 + 4x_3 + x_4 &= 0
\end{aligned}
\]

2. The second system:
\[
\begin{aligned}
x_1 + x_2 + 3x_3 + x_4 &= 3 \\
x_1 - 2x_2 + 3x_3 + x_4 &= 8 \\
x_1 - 5x_2 + 4x_3 + x_4 &= 5
\end{aligned}
\]

You need to use Gaussian elimination to reach row-echelon form and analyze the solutions.

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**C.** The third task asks you to solve two systems of equations using Gaussian elimination:

1. System:
\[
\begin{aligned}
x_1 + 3x_2 + x_3 + x_4 &= 3 \\
2x_1 + 2x_2 + 3x_3 + x_4 &= 8
\end{aligned}
\]

2. System:
\[
\begin{aligned}
3x_1 + 2x_2 + 2x_3 + x_4 &= -1
\end{aligned}
\]

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**D.** The fourth task involves the following homogeneous matrix system and analyzing the rank for different parameter values:

\[
\begin{bmatrix}
1 & 1 & 3 \\
1 & 2 & 3 \\
a & 3 & b
\end{bmatrix}
\]

(a) For which values of \( a \) and \( b \) does the system have infinitely many solutions?  
(b) For which values of \( a \) and \( b \) does the system have no solutions?

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Would you like to solve one of these sections step by step, or should I tackle a specific part first? Let me know!

The image contains a set of matrix and system of equation problems related to linear algebra. Here's a breakdown:

A) Solve a homogeneous system of equations represented by a row-reduced echelon matrix.
B) Use Gaussian elimination to analyze solution sets for two different systems of equations.
C) Solve additional systems using Gaussian elimination and assess their solutions.
D) Analyze the rank of a matrix for different parameter values and determine when the system has infinite or no solutions.

This page covers linear algebra problems involving Gaussian elimination, matrix rank determination, and conditions for infinite or no solutions in systems of equations. Suitable for undergraduate students, the page provides detailed analysis of linear systems and homogeneous matrices.

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