> For the complete documentation index, see [llms.txt](https://tony-ng-1.gitbook.io/matrices-and-linear-algebra-fundamentals/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://tony-ng-1.gitbook.io/matrices-and-linear-algebra-fundamentals/linear-algebra-contd.md).

# Linear Algebra (Pt. 2): 3x3

## 3x3 Linear Equations

It feels I have written a lot, but still have yet to cover **3x3 problems** (fml) 👵.

![oof](https://3978845189-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-LwfxeT-UMPxSGzkwagB%2F-Lwx6ZhMsO__9MbJ6OO_%2F-Lwx6inyCmYebRuy-EMZ%2Fimage.png?alt=media\&token=a90ebc79-c3b9-4504-b077-c8b0a0a05272)

This is also to highlight the *non-linear* increase in complexity to solve these equations with *every* increase in dimension.

In a 3x3 problem, we will require **3 equations** with 3 **variables** (unknowns) to proceed. The following problem is extracted from video lecture 1 of MIT 18.06 on YouTube. 

$$
2x - y = 0\\
-x=2y-z=-1\\
-3y+4z=4
$$

### Row Shorthand

Sticking to the abbreviations used by Strang, the linear equations can be viewed as:

![](https://3978845189-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-LwfxeT-UMPxSGzkwagB%2F-Lwwh9gLJmXt6YT5aSwU%2F-LwwiQ1KsVCQphWfK5I4%2Fimage.png?alt=media\&token=5cc5c955-6b3f-44b2-ace9-81d7f18aefec)

As mentioned previously, the equations for the row picture is left **intact.** Hence, the coefficient matrix A is structured in a way such that each row are the coefficients of each linear equation.

### Row Picture

As now there are 3 unknowns (variables), we would require an **additional axis** and the picture should be a 3-D one.

![Row Picture of 3x3 problem](https://3978845189-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-LwfxeT-UMPxSGzkwagB%2F-LwwkPLjvQYbymqqorpV%2F-Lwx1bqwjtkMrL1qHejA%2Fimage.png?alt=media\&token=fcb2354e-4580-4d03-beb4-4137438e3522)

You will have to excuse my drawing here (despite having a computer lol), as I am not Rembrandt as well 🤪. We can observe that

* **Each equation** has solutions that are essentially a **2-D plane**
* While considering **2 planes**, its simultaneous solution is essentially **a line**
* While considering **3 planes**, its simultaneous solution is essentially **a point**,

all while bearing that each solution to an equation should not be parallel.

As you see, we cannot easily solve for **the point**, the solution that simultaneously satisfy all 3 equations. Thus, the row picture proves to be reliable only up till 2x2 problems.

### Column Shorthand

As mentioned in the column picture, the structure of each equations are broken up into its **columns (variables)**. Therefore, the coefficient matrix will describe each unknown (variable), instead of individual equations. The column shorthand is as of follows:

![Column Shorthand for 3x3](https://3978845189-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-LwfxeT-UMPxSGzkwagB%2F-Lwx7imwH8qWiDUFzhri%2F-LwxBYEU_DAemsfl5HEo%2Fimage.png?alt=media\&token=4ef33eb8-13bc-4d6c-9b84-319dab97e17e)

We know the solution is probably z = 1 as Dr. Strang has specially designed z's coefficient matrix to equate to the result. This is purely illustrating for the purpose of understanding instead of solving (use of elimination method @ next part 👵). 

![Column Picture for 3x3](https://3978845189-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-LwfxeT-UMPxSGzkwagB%2F-Lwx7imwH8qWiDUFzhri%2F-LwxDKN2_biSi9azNQjO%2Fimage.png?alt=media\&token=cb4c09bb-9c14-4414-8efe-19b2bb756448)

Similar to the column picture of 2x2, the coefficient matrices merely represents vectors in 3-D space. Since Col 3 (z) equals to Col 4 (b or RHS), there is no need to go through the whole trouble of finding the linear combination of vectors like 2x2 (thank god 🥵).

#### **More on the next page ⏭**