[Math] Explicit formula for inverse matrix elements

Question

Let $A$ be an $n \times n$ invertible matrix with
\begin{align}
\left(\begin{array}{ccc}
a_{11} & \cdots & a_{1n} \\
\vdots & \ddots & \vdots \\
a_{n1} & \cdots & a_{nn}
\end{array}\right)^{-1}=
\left(\begin{array}{ccc}
b_{11} & \cdots & b_{1n} \\
\vdots & \ddots & \vdots \\
b_{n1} & \cdots & b_{nn}
\end{array}\right)
\end{align}
Is there an explicit formula for $b_{ij}$ in terms of the elements of $A$ and the determinant of $A$?

Edit: Here is a link to the different possible methods to invert a matrix http://en.m.wikipedia.org/wiki/Invertible_matrix#Methods_of_matrix_inversion.

Originally I had in mind using the blockwise inversion method but I think all of the methods require using some submatrix of $A$ and their determinants and / or inverses. I don't think it's possible to write something explicit in terms of $\{a_{ij}\}_{i=1,j=1}^{n,n}$

Answer 1

Hint: $A^{-1}=\frac{1}{|A|}\cdot \text{Adj}(A)$

$\text{Adj}(A)$ is the adjunct matrix of A.