Using data.table in R, can I select a vector of variable columns from a table?

Suppose I have a table like:

> A = data.table(c(1,2,3),c(4,5,6),c(7,8,9))
> A
    V1 V2 V3
1:  1  4  7
2:  2  5  8
3:  3  6  9

I want to select the first column of the first row, second of the second row and third of the third row, getting the result:

> B
[1] 1 5 9

I've tried something like:

B = A[,c(1,2,3)]

But that does not work. Is it possible to do this? (Obviously in my actual use case the column indexes for each row are going to be variables, but I can guarantee a vector of length equal to the number of rows in the table)