Exercise 4

For any given matrix, this theorem tells us that there is a reduced row echelon matrix that is row equivalent to it. This exercise demonstrates why this is the case. Each of the following matrices satisfies three of the four conditions required of a reduced row echelon matrix as prescribed by this definition. For each, indicate how a sequence of row operations can be applied to form a row equivalent reduced row echelon matrix.

1. $$\begin{bmatrix} 1 & 0 & 2 & 3 \\ 0 & 0 & 0 & 0 \\ 0 & 1 & -2 & 1 \\ \end{bmatrix}$$

2. $$\begin{bmatrix} 1 & 0 & 0 & 2 \\ 0 & -2 & 0 & -4 \\ 0 & 0 & 1 & 1 \\ \end{bmatrix}$$

3. $$\begin{bmatrix} 0 & 1 & 0 & -2 \\ 0 & 0 & 1 & 4 \\ 1 & 0 & 0 & 3 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{bmatrix}$$

4. $$\begin{bmatrix} 1 & 0 & 2 & 3 \\ 0 & 1 & 3 & 0 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 \\ \end{bmatrix}$$