Proposition 12

For every invertible square matrix $X$ of type $(n, n)$, the inverse of $X$ is given by the formula

where $Y$ is the cofactor matrix of $X$.

By considering the transpose of $X$ and using Proposition 8 of no. 5, the Laplace expansions could be obtained relative to two complementary sets of rows and, in particular, the expansion of $\det X$ by a row, thus there are formulae equivalent to

in the above notation.

It is easily verified that if $X$ is the matrix of an endomorphism $u$ of a free $\mathbf{A}$-module $M$ of dimension $n$ with respect to a basis $(e_i)_{1 \leq i \leq n}$, ${}^t Y$ is the matrix of the endomorphism $\tilde{u}$ of $M$ defined by the following condition: for every set of $n$ elements $x, y_2, \ldots, y_n$ of $M$,

$\tilde{u}$ is called the cotranspose of $u$ (cf. § 11, no. 11, Corollary to Proposition 13).