Describe explicitly all \(2 \times 2\) row-reduced echelon matrices.
Solution
All \(2\times2\) row-reduced echelon matrices are exactly \[ \begin{bmatrix}0&0\\0&0\end{bmatrix}, \qquad \begin{bmatrix}1&a\\0&0\end{bmatrix}\quad(a\in F), \qquad \begin{bmatrix}0&1\\0&0\end{bmatrix}, \qquad \begin{bmatrix}1&0\\0&1\end{bmatrix}. \]
1. These matrices are all row-reduced echelon matrices.
Each has zero rows below nonzero rows, each leading nonzero entry is \(1\), pivot columns have no other nonzero entries, and successive pivots move strictly to the right.
2. No other \(2\times2\) row-reduced echelon matrix exists.
Classify by rank. Rank \(0\) gives the zero matrix. Rank \(2\) forces pivots in both columns and hence gives \(I_2\). For rank \(1\), there is one nonzero row. If its pivot is in column \(1\), the row is \((1,a)\) with arbitrary \(a\in F\). If its pivot is in column \(2\), its first entry must be zero, giving \((0,1)\). These exhaust all possibilities.