Rank of a Matrix

Definition. Consider two systems

;

.

These systems are said to be linearly dependent if

where k ₁ 0 or k ₂ 0.

If these relations hold only for k ₁ = 0 and k ₂=0, then these two systems are linearly independent.

Consider an matrix

.

Definition. The maximal number of linearly independent rows in the matrix А is called the rank of this matrix.

The maximal number of linearly independent rows equals the maximal number of linearly independent columns.

Definition. A k th-order minor of a matrix А is the determinant consisting of the elements of arbitrarily chosen k columns and k rows.

Theorem. If all k th - order minors of a matrix are zero, then all (k+j)th - order minors are also zero.

Theorem. The rank of a matrix equalsthe maximal order of a nonzero minor.

The first method for calculating the rank of matrix (the bordering method).

(а) The method is to pass lower-order minors to higher-orders minors.

(b) Choose a nonzero minor and border it by a column and a row.

(c) If all of the bordered minors are zero, then the rank of the matrix equals the order of the nonzero k th-order minor:

.

Example. Calculate the rank of a matrix

.

We have , i.e., can the rank not be larger than 4.

,

,

, .

Since all of forth-order minors equal zero and the determinant of third order does not equal zero, it follows that .