The determinant of a matrix $\boldsymbol M$ is so beautiful in part because of the patterns its definition exhibits when generalized to higher dimensions. This can be analyzed by expansion into minors or visually.

The same goes for adding, multiplying and transposing matrices: all three operations, though they may become tedious eventually, are concisely defined, easily recognized, easily applied, and readily generalizable.

The inverse of a matrix, $\boldsymbol{M}^{-1}$, is a different story. The inverse of a $2\times2$ matrix is easy—

$$\begin{bmatrix} a & b \\ c & d \\ \end{bmatrix} ^{-1} = \frac{1}{\det\boldsymbol M} \begin{bmatrix} d & -b \\ -c & a \\ \end{bmatrix}$$

However, the inverse of a $3\times3$ matrix is so verbose that I won’t reproduce here, and from that size up the formulae become incredibly convoluted and include methods such as

  • Newton’s method, $$X_{k+1}=2X_k-X_kAX_k$$
  • the Cayley-Hamilton theorem, which supposedly implies $$\boldsymbol{M}^{-1} = \frac{1}{\det\boldsymbol{M}}\sum_{s=0}^{n-1}\boldsymbol{M}^s\sum_{k_1,k_2,\cdots,k_{n-1}}\prod_{l=1}^{n-1}\frac{(-1)^{l_l+1}}{l^{k_l}k_l!}\operatorname{tr}\left(\boldsymbol{M}^l\right)^{k_l}$$ if $s+\sum_{l=1}^{n-1}k_l=n-1$ for all $k_l\ge0$
  • Eigen decomposition, $$\boldsymbol{M}^{-1} = \boldsymbol{P}\boldsymbol{D}^{-1}\boldsymbol{P}$$ where $\boldsymbol{D}$ is the diagonal matrix of eigenvalues and $\boldsymbol{P}$ is the matrix of eigenvectors

Really, the list just goes on and on!

I was surprised to find Gauss-Jordan elimination on this list, because it is an algorithm that will systematically lead to a conclusion for all matrices. If Gauss-Jordan elimination can be used to find the inverse of an invertible matrix of any size, then why is the hodgepodge of formulae mentioned above so eclectic?

Here’s the real question:

Considering the above information, why are the formulae for the inverses of large matrices all so different?


closed as primarily opinion-based by Carl Christian, Lord Shark the Unknown, Claude Leibovici, mlc, JonMark Perry Nov 4 '17 at 10:04

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