Question

How many types of 2×3 matrices in reduced row-echelon form are there?

Short answer

There are seven types of 2×3 matrices in reduced row-echelon.

Step-by-step solution

Form of 2×3 matrix

All types of 2×3 matrices can be written as .

Where a, b, c, d, e, and f are constants.

2×3 matrix of reduced echelon form

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

Then we have seven options for that

1. At the place of a and f, we have 1. And at b place, we can have y and all other values are 0.

$\left[\begin{array}{ccc}1& y& 0\\ 0& 0& 1\end{array}\right]$

2. At the place of b we have 1. And at the place of c, we have x and all other values are 0.

$\left[\begin{array}{ccc}0& 1& x\\ 0& 0& 0\end{array}\right]$

3. At the place of c, we have 1. And all other values are 0.

$\left[\begin{array}{ccc}0& 0& 1\\ 0& 0& 0\end{array}\right]$

4. At the place of b and f we have 1. And all other values are 0.

$\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\end{array}\right]$

5. All entry of the matrix is 0.

$\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\end{array}\right]$

6.At the place of a and e, we have 1. And all other values are 0.

$\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\end{array}\right]$

7. At the place of a, we have 1. And at the place of c and b, we can have z and k and all other values are 0.

$\left[\begin{array}{ccc}1& z& k\\ 0& 0& 0\end{array}\right]$

Hence, there is total of seven matrices,$\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\end{array}\right],\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\end{array}\right],\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\end{array}\right],\left[\begin{array}{ccc}0& 0& 1\\ 0& 0& 0\end{array}\right],\left[\begin{array}{ccc}0& 1& x\\ 0& 0& 0\end{array}\right],\left[\begin{array}{ccc}1& y& 0\\ 0& 0& 1\end{array}\right],\left[\begin{array}{ccc}1& z& k\\ 0& 0& 0\end{array}\right]$.

Where x, y, z and k are constant.