The difference between the left-handed and right-handed coordinate systems is in orientation of one axis. so in order to transform a matrix from one system to another just multiply it by one of the following matrices, whichever suits you better.
Code:
[-1 0 0 0 ]
[ 0 1 0 0 ]
[ 0 0 1 0 ]
[ 0 0 0 1 ]
[ 1 0 0 0 ]
[ 0 -1 0 0 ]
[ 0 0 1 0 ]
[ 0 0 0 1 ]
[ 1 0 0 0 ]
[ 0 1 0 0 ]
[ 0 0 -1 0 ]
[ 0 0 0 1 ]
It's basically swaping the direction of one axis.
The last one is probably the most common.
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