DF <- matrix(sample(1:9,9),ncol=3,nrow=3)
DF <- as.data.frame.matrix(DF)
> DF
  V1 V2 V3
1  2  7  9
2  8  3  6
3  1  5  4

> DF2
  RE
1 V3
2 V1
3 V2

DF <- data.frame(V1=c(2,8,1),V2=c(7,3,5),V3=c(9,6,4))

colnames(DF)[apply(DF,1,which.max)]
[1] "V3" "V1" "V2"

colnames(DF)[max.col(DF,ties.method="first")]
#[1] "V3" "V1" "V2"

DF <- data.frame(V1=c(2,8,1),V2=c(7,3,5),V3=c(7,6,4))
apply(DF,1,function(x) which(x==max(x)))

[[1]]
V2 V3
 2  3

[[2]]
V1
 1

[[3]]
V2
 2

set.seed(45)
DF <- data.frame(matrix(sample(10, 26746*18, TRUE), ncol=18))

require(data.table)
DT <- data.table(value=unlist(DF, use.names=FALSE),
            colid = 1:nrow(DF), rowid = rep(names(DF), each=nrow(DF)))
setkey(DT, colid, value)
t1 <- DT[J(unique(colid), DT[J(unique(colid)), value, mult="last"]), rowid, mult="first"]

# data.table solution
system.time({
DT <- data.table(value=unlist(DF, use.names=FALSE),
            colid = 1:nrow(DF), rowid = rep(names(DF), each=nrow(DF)))
setkey(DT, colid, value)
t1 <- DT[J(unique(colid), DT[J(unique(colid)), value, mult="last"]), rowid, mult="first"]
})
#   user  system elapsed
#  0.174   0.029   0.227

# apply solution from @thelatemail
system.time(t2 <- colnames(DF)[apply(DF,1,which.max)])
#   user  system elapsed
#  2.322   0.036   2.602

identical(t1, t2)
# [1] TRUE

DT <- data.table(value=unlist(DF, use.names=FALSE),
            colid = 1:nrow(DF), rowid = rep(names(DF), each=nrow(DF)))
setkey(DT, colid, value)
t1 <- DT[J(unique(colid)), rowid, mult="last"]

library(data.table)

set.seed(45)
DT <- data.table(matrix(sample(10, 10^7, TRUE), ncol=10))

system.time(
  DT[, col_max := colnames(.SD)[max.col(.SD, ties.method = "first")]]
)
#>    user  system elapsed
#>    0.15    0.06    0.21
DT[]
#>          V1 V2 V3 V4 V5 V6 V7 V8 V9 V10 col_max
#>       1:  7  4  1  2  3  7  6  6  6   1      V1
#>       2:  4  6  9 10  6  2  7  7  1   3      V4
#>       3:  3  4  9  8  9  9  8  8  6   7      V3
#>       4:  4  8  8  9  7  5  9  2  7   1      V4
#>       5:  4  3  9 10  2  7  9  6  6   9      V4
#>      ---
#>  999996:  4  6 10  5  4  7  3  8  2   8      V3
#>  999997:  8  7  6  6  3 10  2  3 10   1      V6
#>  999998:  2  3  2  7  4  7  5  2  7   3      V4
#>  999999:  8 10  3  2  3  4  5  1  1   4      V2
#> 1000000: 10  4  2  6  6  2  8  4  7   4      V1

DT[, MAX2 := colnames(.SD)[max.col(.SD, ties.method="first")], .SDcols = c("V9", "V10")]

DT[, col_min := colnames(.SD)[max.col(-.SD, ties.method = "first")]]

library(tidyverse)

# sample data frame with a tie
df <- data_frame(V1=c(2,8,1),V2=c(7,3,5),V3=c(9,6,5))

# If you aren't worried about ties:
df %>%
  rownames_to_column('id') %>%  # creates an ID number
  gather(dept, cnt, V1:V3) %>%
  group_by(id) %>%
  slice(which.max(cnt))

# A tibble: 3 x 3
# Groups:   id [3]
  id    dept    cnt
  <chr> <chr> <dbl>
1 1     V3       9.
2 2     V1       8.
3 3     V2       5.


# If you're worried about keeping ties:
df %>%
  rownames_to_column('id') %>%
  gather(dept, cnt, V1:V3) %>%
  group_by(id) %>%
  filter(cnt == max(cnt)) %>% # top_n(cnt, n = 1) also works
  arrange(id)

# A tibble: 4 x 3
# Groups:   id [3]
  id    dept    cnt
  <chr> <chr> <dbl>
1 1     V3       9.
2 2     V1       8.
3 3     V2       5.
4 3     V3       5.


# If you're worried about ties, but only want a certain department, you could use rank() and choose 'first' or 'last'
df %>%
  rownames_to_column('id') %>%
  gather(dept, cnt, V1:V3) %>%
  group_by(id) %>%
  mutate(dept_rank  = rank(-cnt, ties.method = "first")) %>% # or 'last'
  filter(dept_rank == 1) %>%
  select(-dept_rank)

# A tibble: 3 x 3
# Groups:   id [3]
  id    dept    cnt
  <chr> <chr> <dbl>
1 2     V1       8.
2 3     V2       5.
3 1     V3       9.

# if you wanted to keep the original wide data frame
df %>%
  rownames_to_column('id') %>%
  left_join(
    df %>%
      rownames_to_column('id') %>%
      gather(max_dept, max_cnt, V1:V3) %>%
      group_by(id) %>%
      slice(which.max(max_cnt)),
    by = 'id'
  )

# A tibble: 3 x 6
  id       V1    V2    V3 max_dept max_cnt
  <chr> <dbl> <dbl> <dbl> <chr>      <dbl>
1 1        2.    7.    9. V3            9.
2 2        8.    3.    6. V1            8.
3 3        1.    5.    5. V2            5.

DF = data.frame(V1=c(2,8,1),V2=c(7,3,5),V3=c(9,6,4))
DF %>%
  rownames_to_column() %>%
  gather(column, value, -rowname) %>%
  group_by(rowname) %>%
  filter(rank(-value) == 1)

# A tibble: 3 x 3
# Groups:   rowname [3]
  rowname column value
  <chr>   <chr>  <dbl>
1 2       V1         8
2 3       V2         5
3 1       V3         9

DF %>%
  rownames_to_column() %>%
  gather(column, value, -rowname) %>%
  group_by(rowname) %>%
  mutate(rk = rank(-value)) %>%
  filter(rk <= 2) %>%
  arrange(rowname, rk)

# A tibble: 6 x 4
# Groups:   rowname [3]
  rowname column value    rk
  <chr>   <chr>  <dbl> <dbl>
1 1       V3         9     1
2 1       V2         7     2
3 2       V1         8     1
4 2       V3         6     2
5 3       V2         5     1
6 3       V3         4     2

> df<-data.frame(V1=c(2,8,1),V2=c(7,3,5),V3=c(9,6,4))
> df
  V1 V2 V3
1  2  7  9
2  8  3  6
3  1  5  4
> df2<-data.frame()
> for (i in 1:nrow(df)){
+   df2[i,1]<-colnames(df[which.max(df[i,])])
+ }
> df2
  V1
1 V3
2 V1
3 V2

j1 <- max.col(yourDF[, .(V1, V2, V3, V4)], "first")
yourDF$newCol <- c("V1", "V2", "V3", "V4")[j1]