How to find the Inverse of a Matrix

Question

What is a general formula to find the inverse of a matrix? For example, how do you find the inverse of a 2 by 3 or 4 by 4 matrix?

Answer 1

The general formula for a square matrix $A$ is $$A^{-1}=\frac1{\det A}\,\operatorname{com}A,$$ where $\operatorname{com}A$ (the comatrix, or adjugate matrix, of $A$) is the transpose of the matrix of cofactors of $A$.

However, this formula is not used in practice, as its complexity, for a matrix of dimension $n$, is $O(n!)$. A much more efficient method is based on the pivot method: you apply elementary row operations on $A$, until it is transformed into the unit matrix. The same row operations, applied to unit matris, yield $A^{-1}$. This method has complexity $O(n^3)$.

Answer 2

$2\times 3$ matrices doesn't have inverse. And in general, $a\times b$ matrices don't, unless $a = b$.

There are two cute general formulas for $2\times 2$ matrices and $3\times 3$.

$$A = \begin{pmatrix} a & b\\ c & d \end{pmatrix} ~~~~~~~~~~~ A^{-1} = \frac{1}{\text{det}\ A}\begin{pmatrix} d & -b\\ -c & a \end{pmatrix} $$

$$B = \begin{pmatrix} a & b & c\\ d & e & f\\ g & h & i \end{pmatrix} ~~~~~~~~~~~ B^{-1} = \frac{1}{\text{det}\ B} \begin{pmatrix} (ei - fh) & (ch - bi) & (bf - ce)\\ (fg - di) & (ai - cg) & (cd - af)\\ (dh - eg) & (bg - ah) & (ae - bd) \end{pmatrix} $$

For a $4\times 4$ matrix you can use the LaPlace expansion.