Remove rows that have repeated items in a particular column

Dear R Users,

I apology for not being able to provide an adequately informative subject.
Let me describe my problem with an example.

For a matrix
*    (mat <- matrix(c(1,1,2,2,2,3,3, 5,9,1,3,7,4,8), ncol = 2))*
my desired output is
*    (desired <- matrix(c(1,2,3, 5,1,4), ncol = 2))*

That is, the first column is numerically grouped and only the first item in
each group is wanted.  The second column is in increasing order within each
group.  My actual data will be of size 10^6 by 100 so I am hoping to solve
this by a simple function.  Thank you very much for your help.

Best,
Zhongyi Yuan

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