Evaluation of a rank of matrix by a rule of triangle

Scheme:

1) By permutation of rows or columns we choose the element and named it a key element.

2) In following matrix the key element we replace by unit. All elements of row and column at the intersection of which is the key element, we replace by zeros. This row and column we named a key row and key column.

3) All other elements of matrix we will find by a rule of rectangle. If – key element, than the element in a new matrix will be:

.

4) Choose the next key diagonal element and repeat all process.

5) At the end of process we will receive an identity matrix. The order of this matrix will be a rank of the studied matrix.

Remark 6.2: In practice, we usually use a combination of both methods.

§7. 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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