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Matrix Algebra - Rank of a matrix
Rank of a matrix
Let's consider a rectangular matrix:
If to choose in this matrix arbitrarily k rows and k columns, wherethen elements costing on an intersection of these rows and columns, form a square matrix of the k-th order. The determinant of this matrix is called a minor of the k-th order of a matrix A.
The rank of a matrix is the maximal order of a minor of a matrix not equal to zero.
Differently, the rank of a matrix A is equal r , if:
1) there is even one minor of the r-th order of a matrix A, not equal to zero;
2) all minors of the (r +1)-th order and above are equal to zero or do not exist.
Rank of a zero matrix (a matrix consisting of zeros) is considered equal to zero.
The difference min (m, n) – r is called defect of a matrix. If defect of a matrix is equal to zero the matrix has the greatest possible rank.
E x a m p l e . Define a rank of the matrix
S o l u t i o n . The left fourth order minor of the given matrix is equal to
Consequently, the rank of the matrix is equal 4.