Question

We say that two n×mmatrices in reduced row-echelon form are of the same type if they contain the same number of leading 1’s in the same positions. For example,

$$\begin{gathered} \left|120 \\ 001\right| \end{gathered}$$and$$\begin{gathered} \left|130 \\ 001\right| \end{gathered}$$

Are of the same type. How many types of 2×2 matrices in reduced row-echelon form are there?

Short answer

There are four type of 2×2 matrices in reduced row-echelon.

$$\begin{gathered} \left[1 \\ 0\right.\left.c \\ 0\right] \end{gathered}$$Where c is constant. $$\begin{gathered} \left[0 \\ 0\right.\left.1 \\ 0\right],\left[1 \\ 0\right.\left.0 \\ 1\right],\left[0 \\ 0\right.\left.0 \\ 0\right] \end{gathered}$$

Step-by-step solution

Form of 2 × 2 matrix

All type of 2×2 matrix can be written like

$$\begin{gathered} \left[a \\ c\right.\left.b \\ d\right] \end{gathered}$$Where a, b, c, d are constants.

2×2 matrix of reduced echelon form

In the above matrix if we want to reduce it to echelon form

Then we have three options for that

1. The first-row first column entry contains 1 and the rest are 0.

$$\begin{gathered} \left[1 \\ 0\right.\left.c \\ 0\right] \end{gathered}$$

2. The first column contains only 0 and the first-row second entry must be 1.

$$\begin{gathered} \left[0 \\ 0\right.\left.1 \\ 0\right] \end{gathered}$$

3. The first entry of row 1 is1and the second entry is 0. The first entry of the first row is 0 and the second entry is 0.

$$\begin{gathered} \left[1 \\ 0\right.\left.0 \\ 1\right] \end{gathered}$$

4. All entry of the matrix is 0.

$$\begin{gathered} \left[0 \\ 0\right.\left.0 \\ 0\right] \end{gathered}$$

Hence, there are four metrices, that are:$$\begin{gathered} \left[1 \\ 0\right.\left.c \\ 0\right],\left[0 \\ 0\right.\left.1 \\ 0\right],\left[1 \\ 0\right.\left.0 \\ 1\right] \end{gathered}$$ and $$\begin{gathered} \left[0 \\ 0\right.\left.0 \\ 0\right] \end{gathered}$$, Where c is constant.