How do I calculate $e^{tA}$

Question

I want to calculate $e^{tA}$, and eigenvalues are
$\lambda_1=$ trace A , $\lambda_2 = 0 \DeclareMathOperator{\tr}{tr}$
so $P_0=I$ and $P_1 =(A-\lambda_1I)=A-(\tr A)I$
$r_1=e^{(\lambda_1)t} = e^{(\tr A)t}$
$r_2=e^{t(\lambda_2)} \int_0^te^{-s(\lambda_2)}r_{k-1}ds$ = $e^{t*0} \int_0^te^{-s*0}e^{(\tr A)s}ds$
$r_2=1 \int_0^t1*e^{(\tr A)s} = \big[e^{(\tr A)t}-e^{(\tr A)0}\big]$
$r_2=e^{(\tr A)t}-1$

Is this calculation for $r_2$ correct? and how do I calculate $e^{tA}$, i know the formula is
$e^{tA}=r_1(t)P_0+r_2(t)P_1$
$e^{tA}=e^{(\tr A)t}*I+(e^{(\tr A)t}-1)*(A-(\tr A)I)$

But I always make different result in the end, I don't know it is my calculation for $r_2$ is incorrect or i just forgot something?

Thanks a lot for help