To invert one matrix *P*, the various steps are as follows:

- calculate its determinant det(
*P*) - calculate the cofactor
*p*_{ij}of each element, starting from the determinant of the corresponding minor matrix*P*_{{ij}}of*P*, i.e. the matrix*P*, in which one removes the row*i*and the column*j*:

(C.1)

- constitute the cofactor matrix, or the matrix of the cofactors:

(C.2)

- calculate the transpose cofactor matrix
*com*(*P*)*T*, adjugate of*P* - calculate the inverse of
*P*by:

(C.3)

These algebraic calculations are heavy, rather than complex. We will, however, carry them out for the IPM-SM, for inverting the transfer matrix *P* (equation (B.37)); this result is indeed necessary to calculate the control vector.

The following calculations will not detail the intermediate calculation of the coefficients *c**ij* of the adjugate of the *P* matrix, since it is enough to:

- calculate the 3 × 3 determinants of the minor matrices of
*P*, using Cramer’s rules - give to the determinants, the corresponding sign (−1)
^{i + j} - invert the subscripts
*i*and*j*to create the coefficients*cp*_{ij}, of the adjugate matrix:

(C.4)

To calculate the inverse matrix *P*^{− 1}, we will thus calculate firstly the determinant of the transfer ...

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