Matrix Echelon And Reduced Row Echelon Forms For Linear Algebra

When is a matrix in echelon form? Reduced form?

1.) Any and all zero rows are at the bottom.2.) The first non-zero (leading) entry of a row is to the right of any leading entries in the row above it. 3.) Below a leading entry, all entries are zero. ————————————————–4.) The leading entry in each row is 1.5.) Each leading 1 is the only nonzero entry in that column.

A matrix is in echelon form when it satisfies the following three conditions:

1. All rows that contain all zeros are at the bottom.
2. For each row that contains at least one nonzero element, the first nonzero element (called a pivot or a leading coefficient) appears to the right of the pivot of the row above it.
3. The pivots are all 1, and they increase from top to bottom and from left to right.

A matrix is in reduced row echelon form (also called row-reduced echelon form) when it satisfies the following additional condition:

4. In addition to conditions 1–3, the pivot in each nonzero row is the only nonzero entry in its column.

In other words, a matrix in reduced row echelon form is a matrix in echelon form in which all elements below each pivot are zeros, and the pivot in each row is the only nonzero element in its column.

To summarize:
– A matrix is in echelon form if it satisfies conditions 1–3 above.
– A matrix is in reduced row echelon form (or row-reduced echelon form) if it satisfies conditions 1–4 above.

It is important to note that a matrix can have multiple row echelon forms, but only one reduced row echelon form.

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