ediculeN

2021-01-13

Determine which matrices are in reduced echelon form and which others are only in echelon form.
a)$\left[\begin{array}{cccc}1& 0& 1& 0\\ 0& 1& 1& 0\\ 0& 0& 0& 1\end{array}\right]$
b)$\left[\begin{array}{ccccc}0& 1& 1& 1& 1\\ 0& 0& 1& 1& 1\\ 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 0\end{array}\right]$
c)$\left[\begin{array}{cccc}1& 5& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 1\end{array}\right]$

hosentak

Step 1
To Determine which matrices are in reduced echelon form and which others are only in echelon form.
Step 2
Given that
a)$\left[\begin{array}{cccc}1& 0& 1& 0\\ 0& 1& 1& 0\\ 0& 0& 0& 1\end{array}\right]$
According to the definition we conclude that matrix is in reduced row echeclon form because the leading entry in each nonzero row is 1 and each leading 1 is the only nonzero entry in its column.

Given that
b)$\left[\begin{array}{ccccc}0& 1& 1& 1& 1\\ 0& 0& 1& 1& 1\\ 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 0\end{array}\right]$
According to the definition the matrix is in echelon form only.
Echelon form only.
c)Given that
$\left[\begin{array}{cccc}1& 5& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 1\end{array}\right]$
According to the definition we conclude that matrix is not in echelon form and that matrix not in reduced row echelon form.

Jeffrey Jordon

Answer is given below (on video)

xleb123

a) The given matrix is already in reduced echelon form.
$\left[\begin{array}{cccc}1& 0& 1& 0\\ 0& 1& 1& 0\\ 0& 0& 0& 1\end{array}\right]$
b) The given matrix is in echelon form but not in reduced echelon form.
$\left[\begin{array}{ccccc}0& 1& 1& 1& 1\\ 0& 0& 1& 1& 1\\ 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 0\end{array}\right]$
c) The given matrix is in echelon form but not in reduced echelon form.
$\left[\begin{array}{cccc}1& 5& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 1\end{array}\right]$

fudzisako

a) $\left[\begin{array}{cccc}1& 0& 1& 0\\ 0& 1& 1& 0\\ 0& 0& 0& 1\end{array}\right]$
b) $\left[\begin{array}{ccccc}0& 1& 1& 1& 1\\ 0& 0& 1& 1& 1\\ 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 0\end{array}\right]$
c) $\left[\begin{array}{cccc}1& 5& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 1\end{array}\right]$
To determine whether each matrix is in reduced echelon form or only in echelon form, we'll apply the following criteria:
1. Leading coefficient of each row is 1.
2. All entries below each leading coefficient are 0.
3. The leading coefficient of each row is to the right of the leading coefficient of the row above it.
Let's analyze each matrix:
a) $\left[\begin{array}{cccc}1& 0& 1& 0\\ 0& 1& 1& 0\\ 0& 0& 0& 1\end{array}\right]$
This matrix satisfies all the criteria for reduced echelon form. Therefore, it is in reduced echelon form.
b) $\left[\begin{array}{ccccc}0& 1& 1& 1& 1\\ 0& 0& 1& 1& 1\\ 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 0\end{array}\right]$
In this matrix, the leading coefficient of the third row is not to the right of the leading coefficient of the second row. Hence, it does not satisfy the third criterion for reduced echelon form. However, it satisfies the first two criteria for echelon form. Therefore, it is only in echelon form.
c) $\left[\begin{array}{cccc}1& 5& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 1\end{array}\right]$
In this matrix, the leading coefficient of the second row is not 1. Therefore, it does not satisfy the first criterion for reduced echelon form. However, it satisfies the first two criteria for echelon form. Thus, it is only in echelon form.
In summary:
a) The matrix $\left[\begin{array}{cccc}1& 0& 1& 0\\ 0& 1& 1& 0\\ 0& 0& 0& 1\end{array}\right]$ is in reduced echelon form.
b) The matrix $\left[\begin{array}{ccccc}0& 1& 1& 1& 1\\ 0& 0& 1& 1& 1\\ 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 0\end{array}\right]$ is only in echelon form.
c) The matrix $\left[\begin{array}{cccc}1& 5& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 1\end{array}\right]$ is only in echelon form.

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