Inverse of a 3x3 Matrix (Excel Minverse Equivalent)

Introduction

This function will accept one 3x3 matrix as datatable, validate it and give back the inverse as another datatable. I had intentionally used datatable for easy databinding with gridview controls, this is a continuation of Matrix multiplication in my previous tip.

Background

In my new engineering project, I came across the requirement of implementing a lot of matrix arithmetic that was previously done using Excel.

Using the Code

For demonstration purposes, I had created a simple form with a gridview to input 3x3 matrix (please check the image attached).

Then add a Button to the form and on the button click event, send the data of gridview to Calculate Inverse as datatable:

Private Sub Button6_Click(sender As Object, e As EventArgs) Handles Button6.Click
       Dim dt1mat As DataTable = CType((grdMatrix1.DataSource), DataTable)
       grdMatrix3.DataSource = MatrixInverse3x3(dt1mat)
   End Sub

Please find the Inverse function below. Please go through the comments for better understanding.

''' <summary>
   ''' Inverses a Matrix of 3X3 using determinand method and Gaussian elimination
   ''' </summary>
   ''' https://en.wikipedia.org/wiki/Gaussian_elimination#Finding_the_inverse_of_a_matrix
   ''' <param name="dt">Matrix to be inversed</param>
   ''' <returns></returns>
   ''' <remarks></remarks>
   Public Function MatrixInverse3x3(dt As DataTable) As DataTable
 
       Dim dtmyinverse As DataTable = New DataTable
       Dim imyasci As Integer = 65
       For index As Integer = 0 To 2
           dtmyinverse.Columns.Add(Chr(imyasci))
           imyasci += 1
       Next
 
       For index As Integer = 0 To 2
           dtmyinverse.Rows.Add()
       Next
       Dim a As Decimal = Decimal.Parse(dt.Rows(0)(0).ToString())
       Dim b As Decimal = Decimal.Parse(dt.Rows(0)(1).ToString())
       Dim c As Decimal = Decimal.Parse(dt.Rows(0)(2).ToString())
       Dim d As Decimal = Decimal.Parse(dt.Rows(1)(0).ToString())
       Dim e As Decimal = Decimal.Parse(dt.Rows(1)(1).ToString())
       Dim f As Decimal = Decimal.Parse(dt.Rows(1)(2).ToString())
       Dim g As Decimal = Decimal.Parse(dt.Rows(2)(0).ToString())
       Dim h As Decimal = Decimal.Parse(dt.Rows(2)(1).ToString())
       Dim i As Decimal = Decimal.Parse(dt.Rows(2)(2).ToString()) 
 
       'invdet = 1/Sum(v[0]*cross(v[1],v[2])); 
       '         and temp = transpose(mat3(cross(v[1],v[2]),cross(v[2],v[0]),cross(v[0],v[1])))
       ''calculate minor deteminand1
       Dim min1 As Decimal = a * ((e * i) - (f * h))
       ''calculate minor deteminand2
       Dim min2 As Decimal = b * ((d * i) - (f * g))
       ''calculate minor deteminand3
       Dim min3 As Decimal = c * ((d * h) - (e * g))
       ''det=+(minor1) -(minor2)+(minor3)-(minor4)......
       Dim det As Decimal = min1 - min2 + min3
 
 
       'Rule is inverse = 1/det * minor of the TRANSPOSE matrix.  *
       'Note (y,x) becomes (x,y) INTENTIONALLY here!
       det = 1 / det
 
       '        // computes the inverse of a matrix m
       'double det = m(0, 0) * (m(1, 1) * m(2, 2) - m(2, 1) * m(1, 2)) -
       '             m(0, 1) * (m(1, 0) * m(2, 2) - m(1, 2) * m(2, 0)) +
       '             m(0, 2) * (m(1, 0) * m(2, 1) - m(1, 1) * m(2, 0));
 
       dtmyinverse.Rows(0)(0) = (dt.Rows(1)(1) * dt.Rows(2)(2) - dt.Rows(2)(1) * dt.Rows(1)(2)) * det
       dtmyinverse.Rows(0)(1) = (dt.Rows(0)(2) * dt.Rows(2)(1) - dt.Rows(0)(1) * dt.Rows(2)(2)) * det
       dtmyinverse.Rows(0)(2) = (dt.Rows(0)(1) * dt.Rows(1)(2) - dt.Rows(0)(2) * dt.Rows(1)(1)) * det
       dtmyinverse.Rows(1)(0) = (dt.Rows(1)(2) * dt.Rows(2)(0) - dt.Rows(1)(0) * dt.Rows(2)(2)) * det
       dtmyinverse.Rows(1)(1) = (dt.Rows(0)(0) * dt.Rows(2)(2) - dt.Rows(0)(2) * dt.Rows(2)(0)) * det
       dtmyinverse.Rows(1)(2) = (dt.Rows(1)(0) * dt.Rows(0)(2) - dt.Rows(0)(0) * dt.Rows(1)(2)) * det
       dtmyinverse.Rows(2)(0) = (dt.Rows(1)(0) * dt.Rows(2)(1) - dt.Rows(2)(0) * dt.Rows(1)(1)) * det
       dtmyinverse.Rows(2)(1) = (dt.Rows(2)(0) * dt.Rows(0)(1) - dt.Rows(0)(0) * dt.Rows(2)(1)) * det
       dtmyinverse.Rows(2)(2) = (dt.Rows(0)(0) * dt.Rows(1)(1) - dt.Rows(1)(0) * dt.Rows(0)(1)) * det
       'dtmyinverse.Rows(0)(0) = (m(1, 1) * m(2, 2) - m(2, 1) * m(1, 2)) * det
       'dtmyinverse.Rows(0)(1) = (m(0, 2) * m(2, 1) - m(0, 1) * m(2, 2)) * det
       'dtmyinverse.Rows(0)(2) = (m(0, 1) * m(1, 2) - m(0, 2) * m(1, 1)) * det
       'dtmyinverse.Rows(1)(0) = (m(1, 2) * m(2, 0) - m(1, 0) * m(2, 2)) * det
       'dtmyinverse.Rows(1)(1) = (m(0, 0) * m(2, 2) - m(0, 2) * m(2, 0)) * det
       'dtmyinverse.Rows(1)(2) = (m(1, 0) * m(0, 2) - m(0, 0) * m(1, 2)) * det
       'dtmyinverse.Rows(2)(0) = (m(1, 0) * m(2, 1) - m(2, 0) * m(1, 1)) * det
       'dtmyinverse.Rows(2)(1) = (m(2, 0) * m(0, 1) - m(0, 0) * m(2, 1)) * det
       'dtmyinverse.Rows(2)(2) = (m(0, 0) * m(1, 1) - m(1, 0) * m(0, 1)) * det
 
       Return dtmyinverse
   End Function

Points of Interest

It will be interesting to do the same for non 3x3 matrix also and I am currently trying to do the same and will upload once completed.

History

• 23^rd September, 2018: Initial version