Question

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

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

Explanation:

Define echelon form criteria

Using the Matrix Classification knowledge point
A matrix is in echelon form (or row echelon form) if it satisfies three fundamental conditions:

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

Evaluate the given options

We compare each option against the standard definition of echelon form:

• "The matrix has no columns of all zeroes": Incorrect. Echelon form matrices can have columns of all zeroes.
• "Each leading entry of a row is in a column to the right of the leading entry in the row above it": Correct. This is condition 2.
• "Each leading entry is the only nonzero entry in its column": Incorrect. This is a requirement for reduced row echelon form (RREF), not basic echelon form.
• "The leading entry of each row is 1": Incorrect. This is a requirement for reduced row echelon form (RREF). In basic echelon form, leading entries can be any nonzero number.
• "All entries in a column below a leading entry are zeroes": Correct. This is condition 3.
• "Any rows of all zeroes are below any other rows": Correct. This is condition 1.
• "The matrix is in triangular form": Incorrect. Echelon form generalizes triangular form to non-square matrices.

Answer:

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