We are given two matrices:

Matrix $A = \begin{pmatrix} 5 \\ 1 \\ 3 \end{pmatrix}$ (a $3 \times 1$ column vector), and
Matrix $B = \begin{pmatrix} 4 & 1 & 0 \\ 0 & 2 \end{pmatrix}$ (a $2 \times 3$ matrix).

The question asks to determine the result of various operations between $A$ and $B$. These operations involve matrix multiplication, as well as the transposes of these matrices.

Let's break down each option:

a) $A \times B^t$

This means multiplying matrix $A$ with the transpose of $B$, $B^t$.

$B^t = \begin{pmatrix} 4 & 0 \\ 1 & 2 \\ 0 & 2 \end{pmatrix}$ (a $3 \times 2$ matrix).

The multiplication $A \times B^t$ is not defined because $A$ is a $3 \times 1$ matrix and $B^t$ is a $3 \times 2$ matrix. For multiplication to be valid, the number of columns in the first matrix must match the number of rows in the second matrix, which is not the case here.

b) $B \times A^t$

Here, $A^t = \begin{pmatrix} 5 & 1 & 3 \end{pmatrix}$ (a $1 \times 3$ matrix). Matrix $B$ is $2 \times 3$, so the multiplication $B \times A^t$ is valid.

Perform the multiplication: $B \times A^t = \begin{pmatrix} 4 & 1 & 0 \\ 0 & 2 & 2 \end{pmatrix} \times \begin{pmatrix} 5 \\ 1 \\ 3 \end{pmatrix}$