hence, a system of n2 linear algebraic equations in n2 unknowns, i.e. α1,1, α1,2, …, α1,n, α2,1, α2,2, …, α2,n, …, αn,1, αn,2, …, αn,n, arises – the solution of which will be postponed at this point.
If one labels
then Eq. (4.124) will read
in terms of left‐ and right‐hand sides; this is equivalent to Eq. (4.127), as seen above. Assume now that another matrix C exists, such that
thus mimicking the intermediate and right‐hand sides of Eq. (4.124); in view of Eq. (4.64), one has it that
(4.131)
where insertion of Eq. (4.130) unfolds
(4.132)
After applying the associative property as conveyed by Eq. (4.57), one gets
(4.133)
where combination with Eq. (4.129) gives rise to
(4.134)
and finally to
(4.135)
on account of Eqs. (4.61) and (4.128). In other words, the only matrix C that satisfies Eq. (4.130) is indeed A−1, to be calculated via Eq. (4.127), so the second equality in Eq. (4.124) is fully proven once the first equality is true; this result is also consistent with the need that the number of columns of A matches the number of rows of A−1 (thus guaranteeing existence of AA−1) and vice versa (so as to assure existence of A−1 A) – which obviously implies that A and A−1 are square matrices of similar order.
In view of the definition of inverse, one realizes that
based on Eq. (4.124) after replacing A by A−1 – so ordered subtraction of Eq. (4.136) from Eq. (4.124) gives rise to
once the left‐ and middle‐hand sides of the former have been previously swapped; postmultiplication of the first equality in Eq. (4.137) by A produces
(4.138)
where Eqs. (4.70) and (4.76) permit conversion to
The second equality in Eq. (4.124) then supports transformation of Eq. (4.139) to
(4.140)
after having applied the associative property as per Eq. (4.57) – whereas Eq. (4.61) accounts for simplification to
one thus concludes that
after adding ( A−1)−1 to both sides, and recalling Eqs. (4.19) and (4.45). Therefore, composition of the inversion operation with itself cancels it out – in much the same way already found for transposal. A similar reasoning can be developed involving premultiplication of the second equality of Eq. (4.137) by A, viz.
(4.143)
where the distributive property as per Eq.