Solution of system of linear equations in unknowns

This system has the following form:

A matrix is called a basic matrix of the system

.

A matrix, which obtained from matrix by adding column of arbitrary terms of system, is called augmented matrix of the system

.

Definition 7.1: A system having at least one solution is called a compatible system. Otherwise the system is called incompatible.

Theorem 7.1 (Kronecker-Capelli) A system is compatible if and only if the rank of basic matrix is equal to a rank of augmented matrix.

Remark: 1) If the system is compatible and , then the system has an unique solution.

2) If , then the system have infinite number of the solutions.

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