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So let's say that I have this matrix: $$ A=\begin{bmatrix} 2 & 9\\ 3 & 7\\ \end{bmatrix} $$ and the function $f(x)=\frac{1}{1-x}$ and $x\neq{1}$ and I have to determinate the definition domain of the application $t\to f(tA)\in M_2(R)$.

I have tried to solve it with eigenvalues, but I don't realy know if I can continue it. The eigenvalues are: $\frac{9+\sqrt{133}}{2}$ and $\frac{9-\sqrt{133}}{2}$. After this I've tryed to calculate $f(tA)$ and I got $f(tA)=(I-tA)^{-1}$ But I don't know how to calculate this either. Can someone explain me what the definition domain is and how to calculate it?

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    But aren't you done already? You need $I-tA$ to be invertible, which it is for all $t$ except for the eigenvalues. – 2017-01-26
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    @zipirovich I don't actualy understand what the definition domain of an aplication is.. That's what I want to know. A definition for that, I want to understand the exercise.. So can you explain why is that so? – 2017-01-26

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I'm guessing you need to find all $t$ such that $f(tA)$ is well-defined. It's really the usual concept of the domain of a function, where in this particular example the function is $g(t)\colon\mathbb{R}\to M_{2}(\mathbb{R})$ defined as $g(t)=f(tA)=(I-tA)^{-1}$. To be able to evaluate this matrix expression, you need to take the inverse of $I-tA$, so you need $I-tA$ to be invertible. And it is invertible for all $t$ except for the eigenvalues of $A$.