Inverse of an $n\times n$ matrix

Question

I'm curious to know the matrix form of the inverse of an $n\times n$ matrix. Also, how many operations will be needed to compute it?

Answer 1

Like Git Gud said, a closed form would be $$A_{ij}=\frac{(\textrm{adj } A)_{ij}}{\det A}$$ Using the notation $A_{ij}$ for the element at the i-th row and j-th column of $A$.

About the quantity of operations, there is several algorithms for computing $A^{-1}$, and generally they cost $O(n^3)$ operations.

Answer 2

As Integral said, there are many algorithms available for computing matrix inverses. A few of them are described on this Wikipedia page.

Except for very small matrices (of size 2 or 3) encountered in linear algebra homework assignments, no-one would ever compute a matrix inverse using the adjugate/determinant formula. It's complexity is far worse than $O(n^3)$.