Question

which of the following criteria must be satisfied for a matrix to be in reduced echelon form?

all entries in a column below a leading entry are zeroes
each leading entry is the only nonzero entry in its column
each leading entry of a row is in a column to the right of the leading entry in the row above it
any rows of all zeroes are below any other rows
the matrix is in triangular form
the leading entry of each row is 1
the matrix has no columns of all zeroes

Explanation:

Define echelon form criteria

Using the Echelon Form knowledge point
A matrix is in echelon form (or row echelon form) if it satisfies:

1. Any rows of all zeroes are below any other rows.
2. Each leading entry of a row is in a column to the right of the leading entry in the row above it.
3. All entries in a column below a leading entry are zeroes.

Define reduced echelon form criteria

A matrix is in reduced echelon form (or reduced row echelon form) if it satisfies all three echelon form criteria, plus two additional conditions:

1. The leading entry of each nonzero row is 1.
2. Each leading 1 is the only nonzero entry in its column.

Evaluate the given options

• "All entries in a column below a leading entry are zeroes": Required (inherited from echelon form).
• "Each leading entry is the only nonzero entry in its column": Required (specific to reduced echelon form).
• "Each leading entry of a row is in a column to the right of the leading entry in the row above it": Required (inherited from echelon form).
• "Any rows of all zeroes are below any other rows": Required (inherited from echelon form).
• "The matrix is in triangular form": Not required (matrices do not have to be square).
• "The leading entry of each row is 1": Required (specific to reduced echelon form).
• "The matrix has no columns of all zeroes": Not required (columns of all zeroes are allowed).

Answer:

• All entries in a column below a leading entry are zeroes (Correct answer)
• Each leading entry is the only nonzero entry in its column (Correct answer)
• Each leading entry of a row is in a column to the right of the leading entry in the row above it (Correct answer)
• Any rows of all zeroes are below any other rows (Correct answer)
• The matrix is in triangular form
• The leading entry of each row is 1 (Correct answer)
• The matrix has no columns of all zeroes