RM AI Prompt: Question on a Math Problem

### User
How to calculate Lower Triangular Matrix in Cholesky Decomposition?
* Note that the reader is AI learner.

### System
* Your entire response must be contained within a **single markdown code block** (```).
* **Line Breaks**: Strictly use \cr for line breaks instead of \\ for matrcies to avoid WordPress rendering issues,
  while providing LaTex formulas especially matrices.
  (example) $$A = \begin{pmatrix} 1 & 2 \cr 3 & 4 \end{pmatrix}$$
* **Encapsulation**: Always wrap math expressions in `$$...$$` for line-break math or `$...$` for inline math.
   This signals the MathJax/KaTeX renderer to process the content before the WordPress Markdown parser can alter the characters.
* **Subscripts & Underscores**: To prevent WordPress from converting math into italics, escape underscores 
   by using `\_` instead of `_` (e.g., `a\_{ij}`), or omit the underscore entirely (e.g., `aij`)
   if the multi-underscore conflict persists."
* **Subscripts with Asterisks**: To prevent italicization and ensure correct rendering,
   always escape the underscore before an asterisk within curly braces
   (e.g., use `$f\_{\ast}$` or `$x\_{\ast}$` instead of $f_{\ast}$ or $f_*$).
* **Subscripts with Accents**: When a subscript asterisk follows an accent (e.g., \bar or \hat),
   strictly use an escaped underscore within curly braces (e.g., `$\bar{f}\_{\ast}$`) to prevent rendering failure.
* **Vertical Bar in a Table**: When representing the vertical bar character `|` inside a markdown table,
  strictly use the escaped character `\|` to ensure correct table rendering.
* **Code Snippets**: Always wrap code snippets and identifiers in a single pair of single backticks (\`code\`), 
 and strictly avoid using triple backticks for any reason.
* [CRITICAL] Never inject Markdown tags like <em> or <strong> inside math blocks.
    
### Assistant


### Absolute Prohibitions
- Any text outside the code block (greetings, "Understood," "Here is the result," etc.).
- Creating multiple code blocks (all content must be in one single block).

### User
How to calculate Lower Triangular Matrix in Cholesky Decomposition?
* Note that the reader is AI learner.

### System
* Your entire response must be contained within a **single markdown code block** (```).
* **Line Breaks**: Strictly use \cr for line breaks instead of \\ for matrcies to avoid WordPress rendering issues,
  while providing LaTex formulas especially matrices.
  (example) $$A = \begin{pmatrix} 1 & 2 \cr 3 & 4 \end{pmatrix}$$
* **Encapsulation**: Always wrap math expressions in `$$...$$` for line-break math or `$...$` for inline math.
   This signals the MathJax/KaTeX renderer to process the content before the WordPress Markdown parser can alter the characters.
* **Subscripts & Underscores**: To prevent WordPress from converting math into italics, escape underscores 
   by using `\_` instead of `_` (e.g., `a\_{ij}`), or omit the underscore entirely (e.g., `aij`)
   if the multi-underscore conflict persists."
* **Subscripts with Asterisks**: To prevent italicization and ensure correct rendering,
   always escape the underscore before an asterisk within curly braces
   (e.g., use `$f\_{\ast}$` or `$x\_{\ast}$` instead of $f_{\ast}$ or $f_*$).
* **Subscripts with Accents**: When a subscript asterisk follows an accent (e.g., \bar or \hat),
   strictly use an escaped underscore within curly braces (e.g., `$\bar{f}\_{\ast}$`) to prevent rendering failure.
* **Vertical Bar in a Table**: When representing the vertical bar character `|` inside a markdown table,
  strictly use the escaped character `\|` to ensure correct table rendering.
* **Code Snippets**: Always wrap code snippets and identifiers in a single pair of single backticks (\`code\`), 
 and strictly avoid using triple backticks for any reason.
* [CRITICAL] Never inject Markdown tags like <em> or <strong> inside math blocks.
    
### Assistant


### Absolute Prohibitions
- Any text outside the code block (greetings, "Understood," "Here is the result," etc.).
- Creating multiple code blocks (all content must be in one single block).