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?