Вычисление определителя квадратной матрицы с разложением по первой строке

'Ôóíêöèÿ äëÿ èñêëþ÷åíèÿ èç ìàòðèöû arr 1-îé ñòðîêè è j-îãî ñòîëáöà.
'Ðåçóëüòèðóþùàÿ ìàòðèöà èìååò ðàçìåð n x n.
Function Exc1RowJColumn(ByVal arr As Variant, ByVal j As Long, ByVal n As Long) As Variant
    Dim i As Long, jj As Long
    ReDim arr2(1 To n, 1 To n) As Double
    For i = 1 To n
        For jj = 1 To n
            arr2(i, jj) = arr(i + 1, jj + IIf(jj < j, 0, 1))
        Next jj
    Next i
    Exc1RowJColumn = arr2
End Function
'Ðåêóðñèâíàÿ ôóíêöèÿ äëÿ âû÷èñëåíèÿ îïðåäåëèòåëÿ ìàðòèöû arr ðàçìåðîì n x n.
Function Determinant(ByVal arr As Variant, ByVal n As Long) As Double
    Dim j As Long, S As Double
    If n = 1 Then
        Determinant = arr(1, 1) 'Îïðåäåëèòåëü ìàòðèöû ðàçìåðîì 1 x 1 ðàâåí åäèíñòâåííîìó åå ýë-òó.
    Else
        'Âû÷èñëåíèå îïðåäåëèòåëÿ ìàòðèöû n x n ïðè n > 1 ñâîäèòñÿ ê âû÷èñëåíèþ ñóììû ïðîèçâåäåíèé
        'j-îãî ýëåìåíòà ñòðîêè ðàçëîæåíèÿ è îïðåäåëèòåëÿ ìàòðèöû, ïîëó÷åííîé ïóòåì èñêëþ÷åíèÿ èç
        'èñõîäíîé ìàòðèöû âû÷åðêèâàíèåì ñòðîêè ðàçëîæåíèÿ è j-îãî ñòîëáöà. Ñóììà âû÷èñëÿåòñÿ äëÿ
        'j îò 1 äî n. Çíàê ñëàãàåìîãî ðàâåí "+" äëÿ íå÷åòíûõ è "-" äëÿ ÷åòíûõ ñëàãàåìûõ.
        For j = 1 To n
            S = S + (-1) ^ (1 + j) * arr(1, j) * Determinant(Exc1RowJColumn(arr, j, n - 1), n - 1)
        Next j
        Determinant = S
    End If
End Function
'Ïðîâåðêà ôóíêöèè âû÷èñëåíèÿ îïðåäåëèòåëÿ (çàïóñêàòü â Excel).
Sub TestDeterminant()
    Const n = 5
    Dim arr As Variant
    Cells.Clear
    With Cells(1).Resize(n, n)
        .Formula = "=RANDBETWEEN(-10,10)"
        .Value = .Value
        Cells(n + 2, 1).Formula = "=""Îïðåäåëèòåëü, âû÷èñëåííûé ñ ïîìîùüþ âñòðîåííîé ôóíêöèè Excel: ""&MDETERM(" & .Address & ")"
        arr = .Value
    End With
    Cells(n + 4, 1) = "Îïðåäåëèòåëü, âû÷èñëåííûé ñ ïîìîùüþ ôóíêöèè Determinant: " & Determinant(arr, n)
End Sub

'Ôóíêöèÿ äëÿ èñêëþ÷åíèÿ èç ìàòðèöû arr 1-îé ñòðîêè è j-îãî ñòîëáöà.
'Ðåçóëüòèðóþùàÿ ìàòðèöà èìååò ðàçìåð n x n.
Function Exc1RowJColumn(arr As Variant, j As Long, n As Long) As Variant
    Dim i As Long, jj As Long
    ReDim arr2(1 To n, 1 To n) As Double
    For i = 1 To n
        For jj = 1 To n
            If jj < j Then
                arr2(i, jj) = arr(i + 1, jj)
            Else
                arr2(i, jj) = arr(i + 1, jj + 1)
            End If
        Next jj
    Next i
    Exc1RowJColumn = arr2
End Function
'Ðåêóðñèâíàÿ ôóíêöèÿ äëÿ âû÷èñëåíèÿ îïðåäåëèòåëÿ ìàðòèöû arr ðàçìåðîì n x n.
Function Determinant(arr As Variant, n As Long) As Double
    Dim j As Long, S As Double
    If n = 1 Then
        Determinant = arr(1, 1) 'Îïðåäåëèòåëü ìàòðèöû ðàçìåðîì 1 x 1 ðàâåí åäèíñòâåííîìó åå ýë-òó.
    Else
        'Âû÷èñëåíèå îïðåäåëèòåëÿ ìàòðèöû n x n ïðè n > 1 ñâîäèòñÿ ê âû÷èñëåíèþ ñóììû ïðîèçâåäåíèé
        'j-îãî ýëåìåíòà ñòðîêè ðàçëîæåíèÿ è îïðåäåëèòåëÿ ìàòðèöû, ïîëó÷åííîé ïóòåì èñêëþ÷åíèÿ èç
        'èñõîäíîé ìàòðèöû âû÷åðêèâàíèåì ñòðîêè ðàçëîæåíèÿ è j-îãî ñòîëáöà. Ñóììà âû÷èñëÿåòñÿ äëÿ
        'j îò 1 äî n. Çíàê ñëàãàåìîãî ðàâåí "+" äëÿ íå÷åòíûõ è "-" äëÿ ÷åòíûõ ñëàãàåìûõ.
        For j = 1 To n
            S = S + (-1) ^ (1 + j) * arr(1, j) * Determinant(Exc1RowJColumn(arr, j, n - 1), n - 1)
        Next j
        Determinant = S
    End If
End Function
'Ïðîâåðêà ôóíêöèè âû÷èñëåíèÿ îïðåäåëèòåëÿ (çàïóñêàòü â Excel).
Sub TestDeterminant()
    Dim arr As Variant
    If TypeName(Selection) <> "Range" Then
        MsgBox "Âûäåëèòå â êà÷åñòâå ìàòðèöû äèàïàçîí íà ëèñòå."
        Exit Sub
    End If
    arr = Selection.Value
    If Not IsArray(arr) Then
        MsgBox "Îïðåäåëèòåëü âûäåëåííîé ìàòðèöû ðàâåí:   " & arr
        Exit Sub
    End If
    If UBound(arr) <> UBound(arr, 2) Then
        MsgBox "Âûäåëåííàÿ ìàòðèöà äîëæíà áûòü êâàäðàòíîé."
        Exit Sub
    End If
    MsgBox "Îïðåäåëèòåëü âûäåëåííîé ìàòðèöû ðàâåí:   " & Determinant(arr, UBound(arr))
End Sub