Find the rank of `[(3,-1,2),(-3,1,2),(-6,2,4)]`

Explanation

We have
Let `" " A=[(3,-1,2),(-3,1,2),(-6,2,4)]`
Appying `R_(2)to R_(2)+R_(1) and R_(3) to R_(3)+2R_(1),` we get
`A=[(3,-1,2),(0,0,4),(0,0,8)]`
Applying `R_(3) to R_(3)-2R_(2),` we get `A=[(3,-1,2),(0,0,4),(0,0,0)]`
Applying `R_(1) to ((1)/(3)) R_(2) to ((1)/(4))R_(2),` then
`A=[(1,-(1)/(3),(2)/(3)),(0,0,1),(0,0,0)]` this is Echelon form of matrix A.
`therefore " Rank = number of non-zero rows "rArr P(A)=2`
`|A||(3,-1,2),(-3,1.2),(-6,2,4)|`
`=3(4-4)+1(-12+12)+2(-6+6)=0`
`therefore" of A "!= 3` but less than 3.
There will be `""^(3)C_(2)xx""^(3)C_(2)=9` square minors of order 2. Now, we consider of there minors.
(i) `|(1,2),(2,4)|=0` (ii) `|(3,2),(-6,4)|=24!=0`
Hence, all minors are not zero.
Hence, rank of is 2.` rArr p (A)=2`