Summary  
This chapter shows how to implement code that uses matrix multiplication and inversion to solve systems of linear equations, handling warnings for singular or nearly singular matrices and determining unique, infinite, or no solutions via matrix rank checks.  

General domain of usage  
scientific computing

Here you'll learn how to use **matrix multiplication** to solve **systems of equations** that come up in a wide range of **fields**. The general methods can be extrapolated to find solutions to many other **equational forms** (remember the golden rule: if you imagine a good reason to perceive the **transformation** as **linear**, you can use **matrix multiplication** and **Matlab** to solve it!)  

### Task 
Download the program **`Solve_System_Of_Equations`** and follow the instructions at the end of the **video** to modify the variables **`V_vector`** and **`F_vector`** to represent the following system of equations:

$$
\begin{aligned}
3x + 2y + 7z &= 42 \\
17x + 96y + 24z &= 81 \\
-108.5x - 665y - 143.5z &= -420
\end{aligned}
$$

When modifying **`V_vector`**, additional **equations** go in **additional columns**. Additional **variables** go in **new rows** (i.e. each column has an additional row where the new **variable** is represented).  

When modifying **`F_vector`**, the results from additional **equations** go in **additional columns**.  

No changes needed when working with **complex** or **real numbers**: the **transformations** are identical and the **outcome** will automatically correspond to what you input.


Matlab's simplicity, efficiency, and overall computing power make it an excellent programming language for both beginners and top professionals alike. It's the language of choice for anything to do with numbers and data. This is an accelerated course designed to rocket you from beginner to professional, showing you every step along the way, so that you'll walk away ready to begin your programming career. 

Master Matlab's core syntax and coding techniques to create professional-grade programs. Explore essential functions, harness its power for data analysis, modeling, and engineering, and elevate your skills with hands-on exercises and real-world applications that accelerate your learning journey.

Master the foundations of data manipulation in Matlab, including importing data, utilizing for loops, and implementing if statements. Enhance your programming efficiency by modularizing your code, streamlining workflows, and building a solid foundation for advanced Matlab projects.

You'll apply everything you've learned to three real world situations and learn a whole bunch of syntax and details that come up in practice so that you can pull out concrete results from your Matlab programs.  

You'll start by getting your hands dirty exploring Matlab's top-tier visualization and graphing capabilities. After that, you'll learn how to automate graphing and combine the skills you already have to create custom visualizations. And on the way, you'll learn another fundamental aspect of programming: while loops.


To close out, this section will begin by teaching you about recursive programming and how it can help you automate analyses and graphing on a whole new level. After that, you'll dive into the world of matrix multiplication, where you can take advantage of Matlab's state-of-the-art algorithms to open a huge diversity of doors for your programming future. 

Applying Matrix Multiplication: Solving Systems of Equations

Task