Rouchè-Capelli's theorem
Linear systems can be solved by suitably applying Cramer's rule also in more general cases than the one considered in the previous page.
Consider a (rectangular) matrix A. We call minor of order p of A the determinant of a square matrix that can be obtained "intersecting" p rows and p columns (contiguous or not) of A. The entries of the matrix are minors of order 1. A matrix can have many minors of different orders.
Given a matrix A we consider all minors of order one, all minors of order two, and so on. If the two following properties hold:
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at least one minor of order r is different from zero and
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there are no minors of order r+1 or all minors of order r+1 are zero
then the number r is called the rank of the matrix. Obviously r ≤ min(m,n).
Consider a system of m equations in n unknowns, with matrix A and augmented matrix A|b. Then (theorem of Rouchè and Capelli) the system is consistent if and only if A and A|b have the same rank.
This theorem can be proved by applying the Gauss-Jordan algorithm.
As an application we show an alternative method for the resolution of linear systems. When checking the consistency of the system, a minor of order r of the matrix A has been used. Proceed as follows:
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suppress all the equations corresponding to rows that have not been used to obtain the minor;
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transport to the second member the terms containing the unknowns corresponding to columns that have not been used to obtain the minor (so these unknowns are considered like constants);
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the system has now the form of a system of r equations in r unknowns, with non singular matrix; solve it using Cramer's rule.