Editing Rank (linear algebra) (section)

== Applications ==
One useful application of calculating the rank of a matrix is the computation of the number of solutions of a [[system of linear equations]]. According to the [[Rouché–Capelli theorem]], the system is inconsistent if the rank of the [[augmented matrix]] is greater than the rank of the [[coefficient matrix]]. If on the other hand, the ranks of these two matrices are equal, then the system must have at least one solution. The solution is unique if and only if the rank equals the number of variables. Otherwise the general solution has {{mvar|k}} free parameters where {{mvar|k}} is the difference between the number of variables and the rank.  In this case (and assuming the system of equations is in the real or complex numbers) the system of equations has infinitely many solutions.

In [[control theory]], the rank of a matrix can be used to determine whether a [[linear system]] is [[controllability|controllable]], or [[observability|observable]].

In the field of [[communication complexity]], the rank of the communication matrix of a function gives bounds on the amount of communication needed for two parties to compute the function.