[hasch]

here is sagemath

    m = matrix([[0, 1, 0],
                [0, 0, 1],
                [1, 0, 0]])
    v = vector([2, 3, 1])
    m \ v

(1, 2, 3)

and then

    sol = m \ v
    m * sol

(2, 3, 1)