matrices help again (inverse) please

hi there would anyone be able to give me a link or guidence as how i would go about finding the inverse of A? I have a book that tells me add subtract and multiply.

the question is this: find the inverse of A

A =

(19 81)
(2 10)

sorry for the crude matrices. any help would be great :)

lakitu

Quote:

---

Originally Posted by lakitu

hi there would anyone be able to give me a link or guidence as how i would go about finding the inverse of A? I have a book that tells me add subtract and multiply.

the question is this: find the inverse of A

A =

(19 81)
(2 10)

sorry for the crude matrices. any help would be great :)

lakitu

---

Hello.

Matrices can be inverted by various algorithms. It's not easy to explain in a few words so I suggest you Google around. Yot can search for "matrix inversion", "Gauss-Jordan", "LU decomposition", etc.

Or you can go to http://mathworld.wolfram.com/topics/Matrices.html or http://mathworld.wolfram.com/topics/...perations.html or http://mathworld.wolfram.com/MatrixInverse.html

thank you i will try those links now :)

A more general solution

VB Code:

Public Function MxInverse(Matrix() As Double) As Double()
    
    Dim i As Long
    Dim j As Long
    Dim Rows As Long
    Dim Cols As Long
    Dim Tmp() As Double
    Dim Ret() As Double
    Dim Degree As Long
    
    Tmp = Matrix
    
    Rows = UBound(Tmp, 1)
    Cols = UBound(Tmp, 2)
    Degree = Cols + 1
    
    'Augment Identity matrix onto matrix M to get [M|I]
    ReDim Preserve Tmp(Rows, (Degree * 2) - 1)
    For i = Degree To (Degree * 2) - 1
        Tmp(i Mod Degree, i) = 1
    Next
    
    ' Now find the inverse using Gauss-Jordan Elimination which should get us [I|A-1]
    MxGaussJordan Tmp
    
    ' Copy the inverse (A-1) part to array to return
    ReDim Ret(Rows, Cols)
    For i = 0 To Rows
        For j = Degree To (Degree * 2) - 1
            Ret(i, j - Degree) = Tmp(i, j)
        Next
    Next
    
    MxInverse = Ret
    
End Function
 
Public Sub MxGaussJordan(Matrix() As Double)
    
    Dim Rows As Long
    Dim Cols As Long
    Dim P As Long
    Dim i As Long
    Dim j As Long
    Dim m As Double
    Dim d As Double
    Dim Pivot As Double
    
    Rows = UBound(Matrix, 1)
    Cols = UBound(Matrix, 2)
 
    ' Reduce so we get the leading diagonal
    For P = 0 To Rows
        Pivot = Matrix(P, P)
        For i = 0 To Rows
            If Not P = i Then
                m = Matrix(i, P) / Pivot
                For j = 0 To Cols
                    Matrix(i, j) = Matrix(i, j) + (Matrix(P, j) * -m)
                Next
            End If
        Next
    Next
    
    'Divide through to get the identity matrix
    'Note: the identity matrix may have very small values (close to zero)
    'because of the way floating points are stored.
    For i = 0 To Rows
        d = Matrix(i, i)
        For j = 0 To Cols
            Matrix(i, j) = Matrix(i, j) / d
        Next
    Next
    
End Sub

It is not optimised. You should use LU Decomposition for better algorithmic performance, and make of CopyMemory to increase the code performance. Nevertheless; it works

Here is a tutorial with fully worked examples on LU decomposition and its application tfor finding the inverse of a matrix.

And this is from the very famous "Numerical Recipes" book that's available on-line: http://www.library.cornell.edu/nr/bookcpdf/c2-3.pdf

Let Matrix A =
(a b)
(c d)

So the inverse = A^-1 =
(d -b) x 1/[ad-bc]
(-c a)

So for your matrix, the inverse of
(19 81)
(2 10)
is
(10 -81) x 1/(190-162) =
(-2 19)

([5/14] [-81/28])
([-1/14] [19/28]).

Several years ago there was a thread at this forum whihc descibed matrix inversion, determinant evaluation, and solution of simultaneous linear equations.

A search might find it.

Quote:

---

Originally Posted by lakitu

hi there would anyone be able to give me a link or guidence as how i would go about finding the inverse of A? I have a book that tells me add subtract and multiply.

the question is this: find the inverse of A

A =

(19 81)
(2 10)

sorry for the crude matrices. any help would be great :)

lakitu

---

For your martix:

A inverse =

(10 -81)
(-2 19)

You didn't multiply by the reciprocal of the determinant x-ice.