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>> Rank of matrix
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, where
then 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.
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