If A is 3 × 4 matrix and B is a matrix such that

Let $X=\left[x_{i j}\right]_{m \times n}$ and $Y=\left[y_{i j}\right]_{p \times q}$ be two matrices of order $m \times n$ and $p \times q$. The multiplication of matrices $X$ and $Y$ is defined if number of columns of $X$ is same as the

number of rows of $Y$ i.e. $n=p$. Also, $X Y$ is a matrix of order $m \times q$.

The order of matrix $A$ is $3 \times 4$. Therefore, the order of matrix $A^{T}$ is $4 \times 3$.

Let the order of matrix $B$ be $m \times n$.

For $A^{T} B$ to be defined,

$m=3$         (Number of columns of $A^{T}$ is same as the number of rows of $B$ )

For $B A^{\top}$ to be defined,          

$n=4$         (Number of columns of $B$ is same as the number of rows of $A^{T}$ )

Thus, the order of matrix $B$ is $3 \times 4$.

If $A$ is $3 \times 4$ matrix and $B$ is a matrix such that $A^{\top} B$ and $B A^{\top}$ are both defined. Then the order of $B$ is $3 \times 4$