Solution Found!
Determine which matrices are in reduced echelon form and
Chapter 1, Problem 2E(choose chapter or problem)
In Exercises 1 and 2, determine which matrices are in reduced echelon form and which others are only in echelon form.
a. \(\left[\begin{array}{llll}1 & 1 & 0 & 1 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0\end{array}\right]\)
b. \(\left[\begin{array}{llll}1 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1\end{array}\right]\)
c. \(\left[\begin{array}{llll}1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1\end{array}\right]\)
d. \(\left[\begin{array}{lllll}0 & 1 & 1 & 1 & 1 \\ 0 & 0 & 2 & 2 & 2 \\ 0 & 0 & 0 & 0 & 3 \\ 0 & 0 & 0 & 0 & 0\end{array}\right]\)
Questions & Answers
QUESTION:
In Exercises 1 and 2, determine which matrices are in reduced echelon form and which others are only in echelon form.
a. \(\left[\begin{array}{llll}1 & 1 & 0 & 1 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0\end{array}\right]\)
b. \(\left[\begin{array}{llll}1 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1\end{array}\right]\)
c. \(\left[\begin{array}{llll}1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1\end{array}\right]\)
d. \(\left[\begin{array}{lllll}0 & 1 & 1 & 1 & 1 \\ 0 & 0 & 2 & 2 & 2 \\ 0 & 0 & 0 & 0 & 3 \\ 0 & 0 & 0 & 0 & 0\end{array}\right]\)
ANSWER:Answer:
Step 1:
In this problem we need to determine which matrices are in reduced echelon form and which others are only in echelon form.
Definition: A rectangular matrix is in echelon form (or row echelon form ) if it has the following three properties:
- All non zero rows are above any rows of all zeros.
- Each leading entry of a row is in a column to the right of the leading entry of the row above it.
- All entries in a column below a leading entry are zeros.
If a matrix in echelon form satisfies the following additional conditions , then it is in reduced echelon form (or row reduced echelon form):
4) The leading entry in each nonzero row is 1.
5) Each leading 1 is the only nonzero entry in its column.