Show that right multiplication by $A=\left(\begin{array}{ll}a & b \\ c & d\end{array}\right) \in M_{2 \times 2}(\mathbb{C})$ defines a linear transformation $\rho_{A}: M_{2 \times 2}(\mathbb{C}) \rightarrow M_{2 \times 2}(\mathbb{C})$. Find the matrix representing $\rho_{A}$ with respect to the basis

$$\left(\begin{array}{ll} 1 & 0 \\ 0 & 0 \end{array}\right),\left(\begin{array}{ll} 0 & 1 \\ 0 & 0 \end{array}\right),\left(\begin{array}{ll} 0 & 0 \\ 1 & 0 \end{array}\right),\left(\begin{array}{ll} 0 & 0 \\ 0 & 1 \end{array}\right)$$

of $M_{2 \times 2}(\mathbb{C})$. Prove that the characteristic polynomial of $\rho_{A}$ is equal to the square of the characteristic polynomial of $A$, and that $A$ and $\rho_{A}$ have the same minimal polynomial.