The following program tests matrices from 2x2 to 8x8. It finds the 2x2 (not included in the puzzle) has determinant 1, but the ones included in the puzzle (n>4) all have determinant zero.

The determinant evaluation routine was swiped from an old puzzle solution program I had written.  Hopefully it wasn't assuming anything about the matrices whose determinants were being evaluated; the determinants for the 3x3 and 4x4 cases were verified as zero by a TI-84 plus CE calculator, which has a built-in determinant function.

DefDbl A-Z

Dim crlf$, fib(100)

Private Sub Form_Load()

 Form1.Visible = True

 Text1.Text = ""

 crlf = Chr(13) + Chr(10)

 

 maxsize = 10

 a = 1: b = 1

 fib(1) = a: fib(2) = b

 

 For nxt = 3 To 100

   fib(nxt) = fib(nxt - 2) + fib(nxt - 1)

   Text1.Text = Text1.Text & Str(fib(nxt))

 Next

 For sz = 1 To 8

   ReDim triangle(sz, sz)

   For row = 1 To sz

    For col = 1 To sz

      triangle(row, col) = fib((row - 1) * sz + col)

    Next

   Next

   Text1.Text = Text1.Text & sz & Str(det(sz, triangle())) & crlf

   DoEvents

 Next

 

 

 

 Text1.Text = Text1.Text & crlf & " done"

  

End Sub

Function det(sz, triangle())

  If sz = 2 Then

    xx = xx

  End If

  If sz = 1 Then

    det = triangle(1, 1)

    Exit Function

  End If

  ReDim matrix(sz - 1, sz - 1)

  tot = 0

  For col = 1 To sz

    newrow = 0

    For r = 2 To sz

      newrow = newrow + 1

      newcol = 0

      For c = 1 To sz

        If c <> col Then

          newcol = newcol + 1

          matrix(newrow, newcol) = triangle(r, c)

        End If

      Next

    Next r

    term = triangle(1, col) * det(sz - 1, matrix())

    If col Mod 2 = 0 Then term = -term

    tot = tot + term

  Next col

  det = tot

End Function