Linear Algebra Companion: A supplement to undergraduate linear algebra enhanced by computation

While you have learned that every matrix has a unique reduced row-echelon form, many times the RREF has complicated looking entries. This algorithm generates a random matrix whose RREF has integer entries. We have specified the number of non-zero rows in the RREF (=rank) and told it to try to find a matrix whose entries are no larger than 20 in absolute value. Should Sage fail to produce a matrix, increase the upper bound or remove it altogether.

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%display latex

latex.matrix_delimiters("[", "]")

A=random_matrix(QQ,5,7,algorithm='echelonizable', rank=4,

upper_bound=20); A

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A.rref()

You can keep on generating new matrices by clicking the Evaluate (Sage) button. Try to find one where the rref has leading ones which skip a column.