Using the Cayley-Hamilton theorem, show that for a $2 \times 2$ matrix $A$, $$e^A = c_1 A + c_0 I$$ where $c_1$ and $c_2$ are constants.
This problem came to me absolutely from nowhere and I have no clue how to solve it. Basically the theorem tells that every square matrix satisfies its own characteristic equation. By this theorem, we can find the exponent of a square matrix and its inverse. But does it really give a clue how to find anything like $e^A$? Help me to solve this
There are two parts to the question:
Lemma 1 For any polynomial $P,$ $P(A) = c_1(P) A + c_2(P)I.$
Proof sketch The right hand side is equal to the remainder when $P(x)$ is divided by $\chi(x),$ where $\chi$ is the characteristic polynomial of $A.$
Lemma 2 If $P_n$ is the $n$-th Taylor polynomial of $\exp,$ the coefficients $c_{1, 2}(P_n)$ converge as $n$ goes to infinity.
You can go from here.