Characteristic Polynomial linear math

Let $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$
be a matrix of complex numbers. Find the characteristic polynomial $\chi_A(t)$ of $A$ and compute $\chi_A(A)$.

I just wanted to confirm that I did this correctly.

Tha answer I have is:
$$\chi_A(t)= \det\begin{pmatrix}a-t&b\\c&d-t\end{pmatrix}
=(a-t)(d-t)-bc
=ad-bc-at-dt+t^2.

$$
Thus
$$
\chi_A(A)=
\begin{pmatrix}a-(ad-bc-at-dt+t^2)&b\\c&d-(ad-bc-at-dt+t^2)\end{pmatrix}
$$

Is this the right thinking?