2014/04/21 | Improve Society

Inverse of a matrix

If for a given square matrix $A$ there exists a matrix $B$ such that $AB = I$ , then $B$ is called an inverse of $A$ and is denoted by $A^{-1}$ . The following theorem is fundamental.

11. If $A$ is a non-singular square matrix of order n [i.e. $\det(A) \ne 0$ ], then there exists a unique inverse $A^{-1}$ such that $AA^{-1} = A^{-1}A = I$ and we can express $A^{-1}$ in the following form
$\displaystyle A^{-1} = \frac{(A_{jk})^T}{\det(A)} \cdots(14)$
where $(A_{jk})$ is the matrix of cofactors $A_{jk}$ and $(A_{jk})^T = (A_{kj})$ is its transpose.

The following express some properties of the inverse:
$(AB)^{-1} = B^{-1}A^{-1} ,\ (A^{-1})^{-1} = A \cdots(15)$