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I am trying to make a dictionary of matrices. Ideally

MatDic={ 
    matrix(GF(2),2,[1,0,0,1]):'a',
    matrix(GF(2),2,[1,1,0,1]):'b'
    # etc
}

But Sage refuses to hash (mutable) matrices. I then tried

MatDic={ 
    matrix(GF(2),2,[1,0,0,1]).set_immutable():'a',
    matrix(GF(2),2,[1,1,0,1]).set_immutable():'b'
}
MatDic

which gives

{None: 'b'}

I also tried making a dictionary of tuples

MatDic={ 
    (1,0,0,1):'a',
    (1,1,0,1):'b'
}

which raises no errors; However

MatDic[tuple(matrix(GF(2),2,[1,0,0,1]))]

gives

TypeError: mutable vectors are unhashable

What is the best, if possible, way to make a dictionary and lookup matrices?