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Example of row reducing a 4-by-4 matrix

Created over 8 years ago, updated 10 days ago

Example: Row-reduce a 4-by-4 matrix to RREF:

$$A = \begin{pmatrix} 1 & 1 & 1 & 1 \\ 2 & 2 & 3 & 4 \\ 1 & 1 & 2 & 3 \\ 3 & 3 & 4 & 5 \end{pmatrix}$$

Step 1: $R_2 \leftarrow R_2 - 2R_1$, $R_3 \leftarrow R_3 - R_1$, $R_4 \leftarrow R_4 - 3R_1$:
$$\begin{pmatrix} 1 & 1 & 1 & 1 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 1 & 2 \end{pmatrix}$$

Step 2: $R_3 \leftarrow R_3 - R_2$, $R_4 \leftarrow R_4 - R_2$:
$$\begin{pmatrix} 1 & 1 & 1 & 1 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix}$$

Step 3: $R_1 \leftarrow R_1 - R_2$:
$$\begin{pmatrix} 1 & 1 & 0 & -1 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix}$$

Rank = 2. Free variables: $x_2$ and $x_4$.