If two matrices are of the same type and of the same rank, can we deduce that they are equivalent

If two matrices are of the same type and of the same rank, can we deduce that they are equivalent

sure
The necessary and sufficient condition for equivalence of two matrices of the same type is the same rank

If and only if the rank is equal, they have the same canonical form?
It is the textbooks of South China University of technology that have some vague expressions.

Because a and B are of the same order, their canonical form is
Er(A) 0
0 0
and
Er(B) 0
0 0
So if and only if the rank is equal, they have the same canonical form
Note that there is no need for a and B to be equivalent

Let a and B be m × n matrices, and prove that a and B are equivalent if and only if R (a) = R (b)

It is proved that: (necessary) let a and B be equivalent, then B can be regarded as a matrix obtained by a finite elementary transformation, and the elementary transformation does not change the rank of the matrix, so r (a) = R (b). (sufficient) let R (a) = R (b), then the standard forms of a and B are erooo, that is, a and B are equivalent to erooo, so a and B are equivalent