Find inverse of a linear transormation given its matrix

Question

Let $T: R^2 \rightarrow R^2 $ (linear) with matrix $A$:

$$\begin{pmatrix} 3 & 4 \\ 1 & 2 \\ \end{pmatrix} $$

The basis isn't specified. The determinant of $A$ is $2$ , not $0$, so $A$ is invertible and thus $T$ is one-to-one ($T$ is an isomorphism), so there exists $T^{-1}$. How do we find $T^{-1}$?

Answer 1

The formula for the inverse of the matrix $ \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix} $ is $ \frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \\ \end{bmatrix}. $