IIT JEE , AIEEE Forum - Mathematics , Algebra

ganesha1991

can someone explain me

rank of matrix

with an example

chinmay_saxena01

The matrix has a zero determinant and is therefor singular. It has no inverse. If you look the matrix you see that it has two identical rows (and two identical columns). In other words, the rows are not independent. If one row is a multiple of another, then they are not independent, and the determinant is zero. (Equivalently: If one col is a multiple of another, then they are not independent, and the determinant is zero.)

The rank of a matrix is the maximum number of independent rows (or, the maximum number of independent columns). A square matrix An×n is non-singular only if its rank is equal to n.

chinmay_saxena01

and in another way rank of a matrix [x] ,,,,k is if:-
each minor of [x] of order one more than k..i.e. k+1 is zero. and there is at least one minor of order k which doesn't vanish....

chinmay_saxena01

it is nothing but the largest order of any non-vanishing minor of the matrix.....for example .matrix [A] = 1 2 3
2 4 7
3 6 10
here determinant of matrix is zero and there is a minor of order 2 which is not equal to zero ....and it is |2 3|
|4 7| =>14-12 is not equal to zero so the rank of matrix is equal to order of minor i.e. 2

chinmay_saxena01

sorry bro there is a matrix in my post...which is disordered.....and a determinant also....
matrix is:-
1 2 3
2 4 7
3 6 10
and determinant is
2 3
4 7

shub

chinmay has sufficiently explained.....
rank basically is the order of any submatrix (it can include the entire determinent) whose determinent is not equal to 0....

took me a little time to understand it too.....

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