If A is a m × n matrix such that AB and BA are both defined, then B is an

The correct option is "n × m matrix".

To understand why this is correct, let us analyze the conditions under which matrix multiplication is defined.
Recall that for two matrices $X$ and $Y$, the product $XY$ is defined if and only if the number of columns in $X$ is equal to the number of rows in $Y$. If $X$ has dimensions $p\times q$, then $Y$ must have dimensions $q\times r$ for the multiplication $XY$ to be possible.

We are given that $A$ is a matrix of size $m\times n$ (having $m$ rows and $n$ columns). Let the dimensions of matrix $B$ be $x\times y$ (having $x$ rows and $y$ columns).

First, we are told that the product $AB$ is defined.
Since $A$ has dimensions $m\times n$ and $B$ has dimensions $x\times y$, the number of columns of $A$ must equal the number of rows of $B$. Therefore, we must have:
$n=x$

Second, we are told that the product $BA$ is also defined.
Since $B$ has dimensions $x\times y$ and $A$ has dimensions $m\times n$, the number of columns of $B$ must equal the number of rows of $A$. Therefore, we must have:
$y=m$

Combining these two results, we find that matrix $B$ has $x=n$ rows and $y=m$ columns.
Thus, the size of matrix $B$ is $n\times m$.