Show that row equivalence column equivalence and their combi

Show that row equivalence, column equivalence, and their combination are in fact equivalence relations. Find the equivalence classes of the combined row and column relation.

Solution

Two matrices are said to be row equivalent, if both can be reduced to the same form after elementary row operations.

Similarly Two matrices are said to be column equivalent, if both can be reduced to the same form after elementary column operations.

Thus two matrices are said to be equivalent if row equivalent and column equivalent.

Thus combined row and column relation is the relation where two matrices are equivalent if they can be reduced to same combined row and column echelon form.