Linear Algebra: Elimination Method

Purpose: To solve a system of linear equations by transforming the initial matrix $A$ into an upper triangular matrix $(U)$ . It's the standard method used in scientific computing.

Pivots: Key non-zero numbers in the diagonal positions during elimination. They are used to eliminate entries below them.
Multipliers: Numbers used to scale a pivot row before subtracting it from a lower row, aiming to create zeros below the pivot.
Process: For each pivot, calculate the multiplier for the rows below, subtract the scaled pivot row, and create zeros. This process moves from left to right, top to bottom.
Output: The result of successful elimination is an upper triangular matrix $U$ .

Pivots cannot be zero.
The determinant of the original matrix $A$ can be found by multiplying the pivots (e.g., if pivots are $1$ , $2$ , $5$ , the determinant is $1 \times 2 \times 5 = 10$ ).

Failure Condition: A "failure" occurs if a pivot position contains a zero.
Resolution (Row Exchange): If a pivot position is zero, and there is a non-zero entry in the column below it, perform a row exchange (swap the current row with a lower row) to bring a non-zero number into the pivot position.
If all entries below a zero pivot are also zero, elimination cannot proceed in the standard way for that column, indicating potential issues with the system (e.g., singularity if discussing matrix properties).