Here's a stab, with fake data:
set.seed(42)
nc <- 9 ; nr <- 5
mtx <- t(replicate(nr, cumsum(sample(-3:3, size = nc, replace = TRUE))))
dimnames(mtx) <- list(LETTERS[seq_len(nr)], 2000 + seq_len(nc))
mtx
#   2001 2002 2003 2004 2005 2006 2007 2008 2009
# A    3    6    5    7    8    8   10    7    8
# B    1    1    3    6    4    4    7   10    7
# C    0    0    3    0    3    6    3    3    2
# D    3    3    5    7    9    8    9    6    8
# E   -3   -5   -2   -1   -2   -2   -5   -2   -2
So I'm simulating your 175 countries with five letters, and 57 years with 9.
A first cut could be to determine if a year is greater than or equal to the previous year.
t(apply(mtx, 1, diff) >= 0)
#    2002  2003  2004  2005  2006  2007  2008  2009
# A  TRUE FALSE  TRUE  TRUE  TRUE  TRUE FALSE  TRUE
# B  TRUE  TRUE  TRUE FALSE  TRUE  TRUE  TRUE FALSE
# C  TRUE  TRUE FALSE  TRUE  TRUE FALSE  TRUE FALSE
# D  TRUE  TRUE  TRUE  TRUE FALSE  TRUE FALSE  TRUE
# E FALSE  TRUE  TRUE FALSE  TRUE FALSE  TRUE  TRUE
(The need to transpose the result from apply is perhaps unintuitive but a fact of dealing with it. There may be ways to do it without the transpose, but none that I know of in base R that are as direct.)
Note that there are one fewer columns than in the original dataset. I you want to keep track in each year if there is a cumulative increase or not, one might do:
t(apply(mtx, 1, function(a) cumsum(diff(a) >= 0)))
#   2002 2003 2004 2005 2006 2007 2008 2009
# A    1    1    2    3    4    5    5    6
# B    1    2    3    3    4    5    6    6
# C    1    2    2    3    4    4    5    5
# D    1    2    3    4    4    5    5    6
# E    0    1    2    2    3    3    4    5
But I don't necessarily find that more informative. 
From your code, it looks like you might be expecting a single number for each country (letter), in which case you can simplify this further to:
apply(mtx, 1, function(a) sum(diff(a) >= 0))
# A B C D E 
# 6 6 5 6 5 
As far as adding this somehow to your current data, I am neither certain what you want (from your example) nor do I recommend appending summary data to the original matrix. The sole exception is for immediate visualization, not storage.