Question

explain why a matrice can have many row echelon forms

Answer 1

a matrix is in echelon form if it has the shape resulting from a Gaussian elimination. Row echelon form means that Gaussian elimination has operated on the rows and column echelon form means that Gaussian elimination has operated on the columns. In other words, a matrix is in column echelon form if its transpose is in row echelon form. Therefore, only row echelon forms are considered . The similar properties of column echelon form are easily deduced by transposing all the matrices.

Specifically, a matrix is in row echelon form if

• all nonzero rows (rows with at least one nonzero element) are above any rows of all zeroes (all zero rows, if any, belong at the bottom of the matrix), and
• the leading coefficient (the first nonzero number from the left, also called the pivot) of a nonzero row is always strictly to the right of the leading coefficient of the row above it .

These two conditions imply that all entries in a column below a leading coefficient are zeros.