Understand various matrix operations, matrix decompositions, factorization and related operations
Row Echelon Form (REF)
Row echelon form is a simplified matrix form where each leading entry is to the right of the one above, making it ideal for solving linear systems using back-substitution.
A matrix is in Row Echelon Form if:
- All rows of all zeroes are at the bottom.
- The leading entry of each non-zero row is to the right of the leading entry of the row above it.
- All entries below a pivot are zeros.
Input Matrix:
\( A = \begin{bmatrix} 1 & -2 & 1 \\ 1 & -1 & 2 \\ 0 & -2 & 2 \end{bmatrix} \)
Step 1: Eliminate below first pivot
Apply \( R_2 \leftarrow R_2 - R_1 \)
\( \begin{bmatrix} 1 & -2 & 1 \\ 0 & 1 & 1 \\ 0 & -2 & 2 \end{bmatrix} \)
Step 2: Eliminate below pivot in 2nd column
Apply \( R_3 \leftarrow R_3 + 2R_2 \)
\( \begin{bmatrix} 1 & -2 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 4 \end{bmatrix} \)
Final REF:
\( \text{REF}(A) = \begin{bmatrix} 1 & -2 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 4 \end{bmatrix} \)
This matrix is now in row echelon form.