Question

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

Short answer

There are four type of 3×2 matrices in reduced row-echelon.

Step-by-step solution

Form of 3×2 matrix

All type of 3×2 matrix can be written as$\left[\begin{array}{cc}a& b\\ c& d\\ e& f\end{array}\right]$

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

3×2 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. At the place of b we have 1. And all other values are 0.

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

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

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

3. At place of a and d we have 1. And all other values are 0.

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

4. All the values are 0.

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

Hence, there are four types of 3×2 matrices in reduced row-echelon.

$\left[\begin{array}{cc}0& 1\\ 0& 0\\ 0& 0\end{array}\right],\left[\begin{array}{cc}1& y\\ 0& 0\\ 0& x\end{array}\right],\left[\begin{array}{cc}1& 0\\ 0& 1\\ 0& 0\end{array}\right],\left[\begin{array}{cc}0& 0\\ 0& 0\\ 0& 0\end{array}\right]$

Where x and y are constant.