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+open Order
+
+module Order = Order
+
+(*************** Exceptions ***************)
+
+exception NonSquare
+exception ImproperDimensions
+
+(* Functor so we can Abstract away! *)
+module MakeMatrix (C: EltsI.ORDERED_AND_OPERATIONAL) :
+  (MatrixI.MATRIX with type elt = C.t) =
+struct
+
+
+  (*************** End Exceptions ***************)
+
+  (*************** Types ***************)
+
+  type elt = C.t
+
+  (* A matrix is a pair of dimension (n x p) and a array of arrays
+   * the first array is the row (n) and the second the column (p) *)
+  type matrix = (int * int) * (elt array array)
+
+  (*************** End Types ***************)
+
+  (*************** Base Functions ***************)
+
+  (* catching negative dimensions AND 0 dimensions and too large
+   * of a dimension so we don't have to worry about it later *)
+  let empty (rows: int) (columns: int) : matrix =
+    if rows > 0 && columns > 0 then
+      try
+        let m = Array.make_matrix rows columns C.zero in ((rows,columns),m)
+      with _ ->
+        raise ImproperDimensions
+    else (* dimension is negative or 0 *)
+      raise ImproperDimensions
+
+  (*************** End Base Functions ***************)
+
+  (*************** Helper Functions ***************)
+
+  (* get's the nth row of a matrix and returns (r, row) where r is the length
+   * of the row and row is a COPY of the original row. For example, calling
+   * calling get_row m 1 will return (3, |1 3 4 |)
+   *         ________
+   *    m = | 1 3 4 |
+   *        |*2 5 6 |
+  *)
+  (* aside: we don't check whether n < 1 because of our matrix invariant *)
+  let get_row (((n,p),m): matrix) (row: int) : int * elt array =
+    if row <= n then
+      let row' = Array.map (fun x -> x) m.(row - 1) in
+      (p, row')
+    else
+      raise (Failure "Row out of bounds.")
+
+  (* similar to get_row. For m, get_column m 1 will return (2, |1 2|) *)
+  let get_column (((n,p),m): matrix) (column: int) : int * elt array =
+    if column <= p then
+      begin
+        let column' = Array.make n C.zero in
+        for i = 0 to n - 1 do
+          column'.(i) <- m.(i).(column - 1)
+        done;
+        (n, column')
+      end
+    else
+      raise (Failure "Column out of bounds.")
+
+  (* sets the nth row of the matrix m to the specified array a.
+   * This is done IN-PLACE. Therefore the function returns unit.  You should
+   * nonetheless enfore immutability whenever possible.  For a clarification on
+   * what nth row means, look at comment for get_row above. *)
+  let set_row (((n, p), m): matrix) (row: int) (a: elt array) : unit =
+    if row <= n then
+      begin
+        assert(Array.length a = p);
+        for i = 0 to p - 1 do
+          m.(row - 1).(i) <- a.(i)
+        done;
+      end
+    else
+      raise (Failure "Row out of bounds.")
+
+  (* Similar to set_row but sets the nth column instead *)
+  let set_column (((n,p),m): matrix) (column: int) (a: elt array) : unit =
+    if column <= p then
+      begin
+        assert(Array.length a = n);
+        for i = 0 to n - 1 do
+          m.(i).(column - 1) <- a.(i)
+        done;
+      end
+    else
+      raise (Failure "Column out of bounds.")
+
+  (* returns the ij-th element of a matrix (not-zero indexed) *)
+  let get_elt (((n,p),m): matrix) ((i,j): int*int) : elt =
+    if i <= n && j <= p then
+      m.(i - 1).(j - 1)
+    else
+      raise ImproperDimensions
+
+  (* Changes the i,j-th element of a matrix to e. Is not zero-indexed, and
+   * changes the matrix in place *)
+  let set_elt (((n,p),m): matrix) ((i,j): int*int) (e: elt) : unit =
+    if i <= n && j <= p then
+      m.(i - 1).(j - 1) <- e
+    else
+      raise ImproperDimensions
+
+  (* similar to map, but applies to function to the entire matrix
+   * Returns a new matrix *)
+  let map (f: elt -> elt) (mat: matrix) : matrix =
+    let (dim,m) = mat in
+    (dim, Array.map (Array.map f) m)
+
+  (* Just some wrapping of Array.iter made for Matrices! *)
+  let iter (f: elt -> unit) (mat: matrix) : unit =
+    let _, m = mat in
+    Array.iter (Array.iter f) m
+
+  (* Just some wrapping of Array.iteri. Useful for pretty
+   * printing matrix. The index is (i,j). NOT zero-indexed *)
+  let iteri (f: int -> int -> elt -> unit) (mat: matrix) : unit =
+    let _, m = mat in
+    Array.iteri (fun i row -> Array.iteri (fun j e -> f i j e) row) m
+
+  (* folds over each row using base case u and function f *)
+  (* could be a bit more efficient? *)
+  let reduce (f: 'a -> elt -> 'a) (u: 'a) (((p,q),m): matrix) : 'a =
+    let total = ref u in
+    for i = 0 to p - 1 do
+      for j = 0 to q - 1 do
+        total := f (!total) m.(i).(j)
+      done;
+    done;
+    !total
+
+  let fold_row ~(f: elt array -> 'b) ((_,m): matrix) : 'b list =
+
+    let call_row acc v = (f v)::acc in
+    Array.fold_left call_row [] m
+    |> List.rev
+
+
+
+
+  (* given two arrays, this will calculate their dot product *)
+  (* It seems a little funky, but this is done for efficiency's sake.
+   * In short, it tries to multiply each element by it's respective
+   * element until the one array is indexed out of bounds. If the
+   * other array is also out of bounds, then it returns their value.
+   * Otherwise, the arrays were the wrong size and raises ImproperDimension
+
+     THE ABOVE COMMENT HAS NOT BEEN IMPLEMENTED
+
+     Instead we calculate the length before starting
+  *)
+  let dot (v1: elt array) (v2: elt array) : elt =
+    let rec dotting (i: int) (total: elt) : elt =
+      if i = 0 then total
+      else
+        let curr = C.multiply v1.(i-1) v2.(i-1) in
+        dotting (i - 1) (C.add curr total) in
+    let len1, len2 = Array.length v1, Array.length v2 in
+    if len1 = len2 then dotting len1 C.zero
+    else raise ImproperDimensions
+
+  (* function to expose the dimensions of a matrix *)
+  let get_dimensions (m: matrix) : (int * int) =
+    let ((x,y), _) = m in (x,y)
+
+  (*************** End Helper Functions ***************)
+
+
+  (*************** Primary Matrix Functions ***************)
+
+  (* scales a matrix by the appropriate factor *)
+  let scale (m: matrix) (sc: elt) : matrix = map (C.multiply sc) m
+
+  (* Generates a matrix from a list of lists. The inners lists are the rows *)
+  let from_list (lsts : elt list list) : matrix =
+    let check_length (length: int) (lst: elt list) : int =
+      if List.length lst = length then length
+      else raise ImproperDimensions in
+    let p = List.length lsts in
+    match lsts with
+    | [] -> raise ImproperDimensions
+    | hd::tl ->
+      let len = List.length hd in
+      if List.fold_left check_length len tl = len then
+        ((p,len),Array.map Array.of_list (Array.of_list lsts))
+      else
+        raise ImproperDimensions
+
+  (* Generates a matrix from a list of lists. The inners lists are the rows *)
+  let from_array (arrs : elt array array) : matrix =
+    let check_length (length: int) (arr: elt array) : unit =
+      if Array.length arr = length then ()
+      else raise ImproperDimensions in
+    let p = Array.length arrs in
+    match Array.length arrs with
+    | 0 -> raise ImproperDimensions
+    | _ ->
+      let len = Array.length (Array.get arrs 0) in
+      Array.iter (check_length len) arrs;
+      ((p, len), arrs)
+
+  (* Adds two matrices. They must have the same dimensions *)
+  let add ((dim1,m1): matrix) ((dim2,m2): matrix) : matrix =
+    if dim1 = dim2 then
+      let n, p = dim1 in
+      let (dim', sum_m) = empty n p in
+      for i = 0 to n - 1 do
+        for j = 0 to p - 1 do
+          sum_m.(i).(j) <- C.add m1.(i).(j) m2.(i).(j)
+        done;
+      done;
+      (dim',sum_m)
+    else
+      raise ImproperDimensions
+
+
+  (* Multiplies two matrices. If the matrices have dimensions m x n and p x q, n
+   * and p must be equal, and the resulting matrix will have dimension n x q *)
+  let mult (matrix1: matrix) (matrix2: matrix) : matrix =
+    let ((m,n), _), ((p,q), _) = matrix1, matrix2 in
+    if n = p then
+      let (dim, result) = empty m q in
+      for i = 0 to m - 1 do
+        for j = 0 to q - 1 do
+          let (_,row), (_,column) = get_row matrix1 (i + 1),
+                                    get_column matrix2 (j + 1) in
+          result.(i).(j) <- dot row column
+        done;
+      done;
+      (dim,result)
+    else
+      raise ImproperDimensions
+
+  (*************** Helper Functions for Row Reduce ***************)
+
+  (*
+  (* returns the index of the first non-zero elt in an array*)
+  let zero (arr: elt array) : int option =
+    let index = ref 1 in
+    let empty (i: int option) (e: elt) : int option =
+      match i, C.compare e C.zero with
+      | None, Equal -> (index := !index + 1; None)
+      | None, _ -> Some (!index)
+      | _, _ -> i in
+    Array.fold_left empty None arr
+
+  (* returns the the location of the nth non-zero
+   * element in the matrix. Scans column wise.  So the nth non-zero element is
+   * the FIRST non-zero element in the nth non-zero column *)
+  let nth_nz_location (m: matrix) (_: int): (int*int) option =
+    let ((n,p), _) = m in
+    let rec check_col (to_skip: int) (j: int) =
+      if j <= p then
+        let (_,col) = get_column m j in
+        match zero col with
+        | None -> check_col to_skip (j + 1)
+        | Some i ->
+          if to_skip = 0 then
+            Some (i,j)
+          else (* we want a later column *)
+            check_col (to_skip - 1) (j + 1)
+      else None in
+    check_col (n - 1) 1
+
+  (* returns the the location of the first
+   * non-zero and non-one elt. Scans column wise, from
+   * left to right. Basically, it ignores columns
+   * that are all zero or that *)
+  let fst_nz_no_loc (m: matrix): (int*int) option =
+    let ((_, p), _) = m in
+    let rec check_col (j: int) =
+      if j <= p then
+        let (_,col) = get_column m j in
+        match zero col with
+        | None -> check_col (j + 1)
+        | Some i ->
+          match C.compare col.(i-1) C.one with
+          | Equal -> check_col (j + 1)
+          | _ -> Some (i,j)
+      else None in
+    check_col 1
+    *)
+
+  (* Compares two elements in an elt array and returns the greater and its
+   * index. Is a helper function for find_max_col_index *)
+  let compare_helper (e1: elt) (e2: elt) (ind1: int) (ind2: int) : (elt*int) =
+    match C.compare e1 e2 with
+    | Equal -> (e2, ind2)
+    | Greater -> (e1, ind1)
+    | Less -> (e2, ind2)
+
+  (* Finds the element with the greatest absolute value in a column. Is not
+   * 0-indexed. If two elements are both the maximum value, returns the one with
+   * the lowest index. Returns None if this element is zero (if column is all 0)
+  *)
+  let find_max_col_index (array1: elt array) (start_index: int) : int option =
+    let rec find_index (max_index: int) (curr_max: elt) (curr_index: int)
+        (arr: elt array) =
+      if curr_index = Array.length arr then
+        (if curr_max = C.zero then None
+         else Some (max_index+1)) (* Arrays are 0-indexed but matrices aren't *)
+      else
+        (match C.compare arr.(curr_index) C.zero with
+         | Equal -> find_index max_index curr_max (curr_index+1) arr
+         | Greater ->
+           (let (el, index) = compare_helper (arr.(curr_index))
+                curr_max curr_index max_index in
+            find_index index el (curr_index+1) arr)
+         | Less ->
+           (let abs_curr_elt = C.subtract C.zero arr.(curr_index) in
+            let (el, index) = compare_helper abs_curr_elt curr_max curr_index
+                max_index in
+            find_index index el (curr_index+1) arr))
+    in
+    find_index 0 C.zero (start_index -1) array1
+
+  (* Basic row operations *)
+  (* Scales a row by sc *)
+  let scale_row (m: matrix) (num: int) (sc: elt) : unit =
+    let (_, row) = get_row m num in
+    let new_row = Array.map (C.multiply sc) row in
+    set_row m num new_row
+
+  (* Swaps two rows of a matrix *)
+  let swap_row (m: matrix) (r1: int) (r2: int) : unit =
+    let (len1, row1) = get_row m r1 in
+    let (len2, row2) = get_row m r2 in
+    let _ = assert (len1 = len2) in
+    let _ = set_row m r1 row2 in
+    let _ = set_row m r2 row1 in
+    ()
+
+  (* Subtracts a multiple of r2 from r1 *)
+  let sub_mult (m: matrix) (r1: int) (r2: int) (sc: elt) : unit =
+    let (len1, row1) = get_row m r1 in
+    let (len2, row2) = get_row m r2 in
+    let _ = assert (len1 = len2) in
+    for i = 0 to len1 - 1 do (* Arrays are 0-indexed *)
+      row1.(i) <- C.subtract row1.(i) (C.multiply sc row2.(i))
+    done;
+    set_row m r1 row1
+
+  (*************** End Helper Functions for Row Reduce ***************)
+
+  (* Returns the row reduced form of a matrix. Is not done in place, but creates
+   * a new matrix *)
+  let row_reduce (mat: matrix) : matrix =
+    let[@tailcall] rec row_reduce_h (n_row: int) (n_col: int) (mat2: matrix) : unit =
+      let ((num_row, _), _) = mat2 in
+      if (n_col = num_row + 1) then ()
+      else
+        let (_,col) = get_column mat2 n_col in
+        match find_max_col_index col n_row with
+        | None (* Column all 0s *) -> row_reduce_h n_row (n_col+1) mat2
+        | Some index ->
+          begin
+            swap_row mat2 index n_row;
+            let pivot = get_elt mat2 (n_row, n_col) in
+            scale_row mat2 (n_row) (C.divide C.one pivot);
+            for i = 1 to num_row do
+              if i <> n_row then sub_mult mat2 i n_row (get_elt mat2 (i,n_col))
+            done;
+            row_reduce_h (n_row+1) (n_col+1) mat2
+          end
+    in
+    (* Copies the matrix *)
+    let ((n,p),m) = mat in
+    let (dim,mat_cp) = empty n p in
+    for i = 0 to n - 1 do
+      for j = 0 to p - 1 do
+        mat_cp.(i).(j) <- m.(i).(j)
+      done;
+    done;
+    let _ = row_reduce_h 1 1 (dim,mat_cp) in (dim,mat_cp)
+
+  (*************** End Main Functions ***************)
+
+  (*************** Optional module functions ***************)
+
+  (* calculates the trace of a matrix *)
+  let trace (((n,p),m): matrix) : elt =
+    let rec build (elt: elt) (i: int) =
+      if i > -1 then
+        build (C.add m.(i).(i) elt) (i - 1)
+      else
+        elt in
+    if n = p then build C.zero (n - 1)
+    else raise ImproperDimensions
+
+  (* calculates the transpose of a matrix and retuns a new one *)
+  let transpose (((n,p),m): matrix) =
+    let (dim,m') = empty p n in
+    for i = 0 to n - 1 do
+      for j = 0 to p - 1 do
+        m'.(j).(i) <- m.(i).(j)
+      done;
+    done;
+    assert(dim = (p,n));
+    ((p,n),m')
+
+  (* Returns the inverse of a matrix. Uses a pretty simple algorithm *)
+  let inverse (mat: matrix) : matrix =
+    let ((n, p), _) = mat in
+    if n = p then
+      (* create augmented matrix *)
+      let augmented = empty n (2*n) in
+      for i = 1 to n do
+        let (dim,col) = get_column mat i in
+        let arr = Array.make n C.zero in
+        begin
+          assert(dim = n);
+          arr.(i-1) <- C.one;
+          set_column augmented i col;
+          set_column augmented (n + i) arr
+        end
+      done;
+      let augmented' = row_reduce augmented in
+      (* create the inverted matrix and fill in with appropriate values *)
+      let inverse = empty n n in
+      for i = 1 to n do
+        let (dim, col) = get_column augmented' (n + i) in
+        let _ = assert(dim = n) in
+        let _ = set_column inverse i col in
+        ()
+      done;
+      inverse
+    else
+      raise NonSquare
+
+  (***************** HELPER FUNCTIONS FOR DETERMINANT *****************)
+  (* creates an identity matrix of size n*)
+  let create_identity (n:int) : matrix =
+    let (dim,m) = empty n n in
+    for i = 0 to n - 1 do
+      m.(i).(i) <- C.one
+    done;
+    (dim,m)
+
+  (* Finds the index of the maximum value of an array *)
+  let find_max_index (arr: elt array) (start_index : int) : int =
+    let rec find_index (max_index: int) (curr_index: int) =
+      if curr_index = Array.length arr then max_index+1
+      else
+        match C.compare arr.(curr_index) arr.(max_index) with
+        | Equal | Less -> find_index max_index (curr_index + 1)
+        | Greater -> find_index curr_index (curr_index + 1) in
+    find_index (start_index - 1) start_index
+
+  (* Creates the pivoting matrix for A. Returns swqps. Adapted from
+   * http://rosettacode.org/wiki/LU_decomposition#Common_Lisp *)
+  let pivotize (((n,p),m): matrix) : matrix * int =
+    if n = p then
+      let swaps = ref 0 in
+      let pivot_mat = create_identity n in
+      for j = 1 to n do
+        let (_,col) = get_column ((n,p),m) j in
+        let max_index = find_max_index col j in
+        if max_index <> j then
+          (swaps := !swaps + 1; swap_row pivot_mat max_index j)
+        else ()
+      done;
+      (pivot_mat,!swaps)
+    else raise ImproperDimensions
+
+  (* decomposes a matrix into a lower triangualar, upper triangualar
+   * and a pivot matrix. It returns (L,U,P). Adapted from
+   * http://rosettacode.org/wiki/LU_decomposition#Common_Lisp *)
+  let lu_decomposition (((n,p),m): matrix) : (matrix*matrix*matrix)*int =
+    if n = p then
+      let mat = ((n,p),m) in
+      let lower, upper, (pivot,s) = empty n n, empty n n, pivotize mat in
+      let (_ ,l),(_ ,u), _ = lower,upper,pivot in
+      let ((_, _),mat') = mult pivot mat in
+      for j = 0 to n - 1 do
+        l.(j).(j) <- C.one;
+        for i = 0 to j do
+          let sum = ref C.zero in
+          for k = 0 to i - 1 do
+            sum := C.add (!sum) (C.multiply u.(k).(j) l.(i).(k))
+          done;
+          u.(i).(j) <- C.subtract mat'.(i).(j) (!sum)
+        done;
+        for i = j to n - 1 do
+          let sum = ref C.zero in
+          for k = 0 to j - 1 do
+            sum := C.add (!sum) (C.multiply u.(k).(j) l.(i).(k))
+          done;
+          let sub = C.subtract mat'.(i).(j) (!sum) in
+          l.(i).(j) <- C.divide sub u.(j).(j)
+        done;
+      done;
+      (lower,upper,pivot),s
+    else raise ImproperDimensions
+
+  (* Computes the determinant of a matrix *)
+  let determinant (m: matrix) : elt =
+    try
+      let ((n,p), _) = m in
+      if n = p then
+        let rec triangualar_det (a,mat) curr_index acc =
+          if curr_index < n then
+            let acc' = C.multiply mat.(curr_index).(curr_index) acc in
+            triangualar_det (a,mat) (curr_index + 1) acc'
+          else acc in
+        let ((dim1,l),(dim2,u), _),s = lu_decomposition m in
+        let det1, det2 = triangualar_det (dim1,l) 0 C.one,
+                         triangualar_det (dim2,u) 0 C.one in
+        if s mod 2 = 0 then C.multiply det1 det2
+        else C.subtract C.zero (C.multiply det1 det2)
+      else raise ImproperDimensions
+    with
+    | _ -> C.zero
+
+
+  (*************** Optional module functions ***************)
+
+
+end