Use row and columns indices in matrix to extract values from matrix

Question

I have a Matrix A which looks like

A = matrix(1:9,3,3)
A
     [,1] [,2] [,3]
[1,]    1    4    7
[2,]    2    5    8
[3,]    3    6    9

and a matrix of indices of elements I am interested in. Column 1 contains row indices, and column 2 contains column indices:

v = matrix(c(1, 3, 2, 2, 2, 3), nrow = 3, ncol = 2)
v
     [,1] [,2]
[1,]    1    2
[2,]    3    2
[3,]    2    3

I want to use the rows and column indices in 'v' to extract numbers from 'A'; the indices correspond to the numbers 4 (A[1, 2]), 6 (A[3, 2]) and 8.

How can I extract those numbers directly from 'A' without using a loop?

When I use

A[v[ , 1], v[ , 2]]

I get

     [,1] [,2] [,3]
[1,]    4    4    7
[2,]    6    6    9
[3,]    5    5    8

because R takes all combinations of the first and second column of 'v'.

What I want is an expression which gives me directly 4, 6, 8.

I could just take the diagonal elements but there must be an easier way.

Answer 1

From ?"[", you will find the following:

When indexing arrays by [ a single argument i can be a matrix with as many columns as there are dimensions of x; the result is then a vector with elements corresponding to the sets of indices in each row of i.

and later on...

A third form of indexing is via a numeric matrix with the one column for each dimension: each row of the index matrix then selects a single element of the array, and the result is a vector. Negative indices are not allowed in the index matrix. NA and zero values are allowed: rows of an index matrix containing a zero are ignored, whereas rows containing an NA produce an NA in the result.

Thus, what you are looking for is simply:

A[v]