Gaussian Elimination:

As part of learning from Gilbert Strang's amazing "Linear Algebra and Its Applications", I've implemented Gaussian Elimination implementation.

Algorithm Steps

An augmented matrix $A$ of size $n \times (n+1)$ representing the system of linear equations $Ax = b$.

Solution vector $x$ if the system has a unique solution, or a notification of a singular or inconsistent system.

Forward Elimination: 1.1. For each row $i$ from $0$ to $n-1$:
- 1.1.1. Pivot Selection (Partial Pivoting):
  - Find the row $k$ with the maximum absolute value in column $i$ below or at the diagonal (i.e., among $i$ to $n-1$).
  - Swap row $i$ with row $k$ to bring the largest absolute value to the pivot position.
- 1.1.2. Pivot Validation:
  - If the pivot element $A[i][i]$ is zero after swapping, the matrix is singular (no unique solution).
- 1.1.3. Elimination:
  - For each row $j$ below row $i$ (i.e., $j = i+1$ to $n-1$):
  - Compute the elimination factor $f = \frac{A[j][i]}{A[i][i]}$.
  - For each column $k$ from $i$ to $n$ (including the RHS):
  - Update $A[j][k] = A[j][k] - f \cdot A[i][k]$.
Back Substitution:
- 2.1. Initialize a solution vector $x$ of size $n$.
- 2.2. For each row $i$ from $n-1$ to $0$ (in reverse order):
  - 2.2.1. Compute the sum of known values:
    - $\text{sum} = \sum_{j=i+1}^{n-1} A[i][j] \cdot x[j]$.
  - 2.2.2. Compute the value of the unknown $x[i]$:
    - $x[i] = \frac{A[i][n] - \text{sum}}{A[i][i]}$ .
Result:
- The vector $x$ contains the solutions to the system of linear equations $Ax = b$.