If both AB and BA are defined, must A and B be square matrices?

[Bashyboy]

Homework Statement

What do we know about the sizes of the matrices A and
B if both of the products AB and BA are defined?

Homework Equations

The Attempt at a Solution

I determined the sizes, of which were correct; but I was wondering if A and B would always be squares. Well, would they always be squares?

 

[clamtrox]

Well, when is the product AB defined? How about BA?

 

[Mark44]

Homework Statement

What do we know about the sizes of the matrices A and
B if both of the products AB and BA are defined?

Homework Equations

The Attempt at a Solution

I determined the sizes, of which were correct; but I was wondering if A and B would always be squares. Well, would they always be squares?

Suppose A is m X n (m rows and n columns), and B is r X s. For AB to be defined, what relationship must there be between A's rows or columns and B's rows or columns? How many rows and columns will AB have?

For BA to be defined, what relationship must there be between B's rows or columns and A's rows or columns? How many rows and columns will BA have?

 

[Bashyboy]

For AB to be defined, n = r. For BA to be defined, s = m. Right?

 

[Mark44]

For AB to be defined, n = r. For BA to be defined, s = m. Right?

Right.