Det.hs

import Data.Array
import System.Random
import Data.Ratio ((%), Ratio)

type MatrixIndex = (Int, Int)
type MatrixElement = Rational
type MatrixValue = Array MatrixIndex MatrixElement
newtype MatrixDimension = MatrixDimension Int deriving (Show)

data Matrix = Matrix
            { n   :: !MatrixDimension
            , mat :: !MatrixValue
            } deriving (Show)

randomMatrix :: MatrixDimension -> IO Matrix
randomMatrix n@(MatrixDimension n') = Matrix n <$> mat
  where
    mat :: IO MatrixValue
    mat = listArray idx <$> values
    idx :: (MatrixIndex, MatrixIndex)
    idx = ((1, 1), (n', n'))
    values = take (n' ^ 2) <$> randomNumbers

randomNumbers :: IO [Rational]
randomNumbers = do
  g <- getStdGen
  return $ map ratio $ randomRs (1, 9) g
    where 
      ratio n = n % 1

det :: Matrix -> Rational
det m = go 1 m
  where
    go a m'@(Matrix (MatrixDimension 1) mat) = a * leftTop m'
    go a m'@(Matrix (MatrixDimension n) mat) = go (a * leftTop m') (reduceDimension m')

leftTop :: Matrix -> Rational
leftTop (Matrix _ mat )= mat ! (1, 1)

reduceDimension :: Matrix -> Matrix
reduceDimension m@(Matrix (MatrixDimension n) _) = Matrix (MatrixDimension (n-1)) mat
  where
    mat = leftZero $ normalizeTop m

normalizeTop :: Matrix -> Matrix
normalizeTop m@(Matrix dim@(MatrixDimension n) mat) = Matrix dim mat'
  where
    mat' = accum (/) mat $ [((1, x), lt) | x <- [1..n]]
    lt = leftTop m

leftZero :: Matrix -> MatrixValue
leftZero (Matrix (MatrixDimension n) mat) = listArray ((1, 1) ,(n', n')) $ zipWith f xs ys
  where
    n' = n - 1
    f (a, x) y = x - a * y
    xs = [(mat ! (row, 1), mat ! (row, col)) | row <- [2..n], col <- [2..n]]
    ys = concat $ take n' $ repeat [mat ! (1, col) | col <- [2..n]]

leftZero' (Matrix (MatrixDimension n) mat) = ys
  where
    n' = n - 1
    f (a, x) y = x - a * y
    xs = [(mat ! (row, 1), mat ! (row, col)) | row <- [2..n], col <- [2..n]]
    ys = concat $ repeat [mat ! (1, col) | col <- [2..n]]
main :: IO ()
main = do
  ns <- randomMatrix $ MatrixDimension 2
  print ""
  print ns
  print $ det ns
--   print $ normalizeTop ns
--   print $ leftZero $ normalizeTop ns

  print "Det Matrix"