Simple 3x3 matrix inverse code C

Question

What s the easiest way to compute a 3x3 matrix inverse   I m just looking for a short code snippet that ll do the trick for non-singular matrices  possibly using Cramer s rule   It doesn t need to be highly optimized  I d prefer simplicity over speed  I d rather not link in additional libraries

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This piece of code computes the transposed inverse of the matrix A    double determinant       A 0 0   A 1 1  A 2 2 -A 2 1  A 1 2                           -A 0 1   A 1 0  A 2 2 -A 1 2  A 2 0                            A 0 2   A 1 0  A 2 1 -A 1 1  A 2 0    double invdet   1 determinant  result 0 0      A 1 1  A 2 2 -A 2 1  A 1 2   invdet  result 1 0    - A 0 1  A 2 2 -A 0 2  A 2 1   invdet  result 2 0      A 0 1  A 1 2 -A 0 2  A 1 1   invdet  result 0 1    - A 1 0  A 2 2 -A 1 2  A 2 0   invdet  result 1 1      A 0 0  A 2 2 -A 0 2  A 2 0   invdet  result 2 1    - A 0 0  A 1 2 -A 1 0  A 0 2   invdet  result 0 2      A 1 0  A 2 1 -A 2 0  A 1 1   invdet  result 1 2    - A 0 0  A 2 1 -A 2 0  A 0 1   invdet  result 2 2      A 0 0  A 1 1 -A 1 0  A 0 1   invdet    Though the question stipulated non-singular matrices  you might still want to check if determinant equals zero  or very near zero  and flag it in some way to be safe

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I would also recommend Ilmbase  which is part of OpenEXR   It s a good set of templated 2 3 4-vector and matrix routines

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Title  Matrix Header File   Writer  Say OL   This is a beginner code not an expert one   No responsibilty for any errors   Use for your own risk using namespace std  int row col Row Col  double Coefficient    Input Matrix void Input double Matrix 9  9  int Row int Col        for row 1 row lt  Row row            for col 1 col lt  Col col                          cout lt  lt  e   lt  lt row lt  lt      lt  lt col lt  lt                   cin gt  gt Matrix row  col                 Output Matrix void Output double Matrix 9  9  int Row int Col        for row 1 row lt  Row row                  for col 1 col lt  Col col                cout lt  lt Matrix row  col  lt  lt   t           cout lt  lt endl            Copy Pointer to Matrix void CopyPointer double   Pointer  9  double Matrix 9  9  int Row int Col        for row 1 row lt  Row row            for col 1 col lt  Col col                Matrix row  col  Pointer row  col       Copy Matrix to Matrix void CopyMatrix double MatrixInput 9  9  double MatrixTarget 9  9  int Row int Col        for row 1 row lt  Row row            for col 1 col lt  Col col                MatrixTarget row  col  MatrixInput row  col       Transpose of Matrix double MatrixTran 9  9   double    Transpose  double MatrixInput 9  9  int Row int Col   9        for row 1 row lt  Row row            for col 1 col lt  Col col                MatrixTran col  row  MatrixInput row  col       return MatrixTran      Matrix Addition double MatrixAdd 9  9   double    Addition  double MatrixA 9  9  double MatrixB 9  9  int Row int Col   9        for row 1 row lt  Row row            for col 1 col lt  Col col                MatrixAdd row  col  MatrixA row  col  MatrixB row  col       return MatrixAdd      Matrix Subtraction double MatrixSub 9  9   double    Subtraction  double MatrixA 9  9  double MatrixB 9  9  int Row int Col   9        for row 1 row lt  Row row            for col 1 col lt  Col col                MatrixSub row  col  MatrixA row  col -MatrixB row  col       return MatrixSub      Matrix Multiplication int mRow nCol pCol kcol  double MatrixMult 9  9   double    Multiplication  double MatrixA 9  9  double MatrixB 9  9  int mRow int nCol int pCol   9        for row 1 row lt  mRow row            for col 1 col lt  pCol col                          MatrixMult row  col  0 0              for kcol 1 kcol lt  nCol kcol                    MatrixMult row  col   MatrixA row  kcol  MatrixB kcol  col                 return MatrixMult      Interchange Two Rows double RowTemp 9  9   double MatrixInter 9  9   double    InterchangeRow  double MatrixInput 9  9  int Row int Col int iRow int jRow   9        CopyMatrix MatrixInput MatrixInter Row Col       for col 1 col lt  Col col                  RowTemp iRow  col  MatrixInter iRow  col           MatrixInter iRow  col  MatrixInter jRow  col           MatrixInter jRow  col  RowTemp iRow  col             return MatrixInter      Pivote Downward double MatrixDown 9  9   double    PivoteDown  double MatrixInput 9  9  int Row int Col int tRow int tCol   9        CopyMatrix MatrixInput MatrixDown Row Col       Coefficient MatrixDown tRow  tCol       if Coefficient  1 0          for col 1 col lt  Col col                MatrixDown tRow  col   Coefficient      if tRow lt Row          for row tRow 1 row lt  Row row                          Coefficient MatrixDown row  tCol               for col 1 col lt  Col col                    MatrixDown row  col - Coefficient MatrixDown tRow  col             return MatrixDown      Pivote Upward double MatrixUp 9  9   double    PivoteUp  double MatrixInput 9  9  int Row int Col int tRow int tCol   9        CopyMatrix MatrixInput MatrixUp Row Col       Coefficient MatrixUp tRow  tCol       if Coefficient  1 0          for col 1 col lt  Col col                MatrixUp tRow  col   Coefficient      if tRow gt 1          for row tRow-1 row gt  1 row--                        Coefficient MatrixUp row  tCol               for col 1 col lt  Col col                    MatrixUp row  col - Coefficient MatrixUp tRow  col                 return MatrixUp      Pivote in Determinant double MatrixPiv 9  9   double    Pivote  double MatrixInput 9  9  int Dim int pTarget   9        CopyMatrix MatrixInput MatrixPiv Dim Dim       for row pTarget 1 row lt  Dim row                  Coefficient MatrixPiv row  pTarget  MatrixPiv pTarget  pTarget           for col 1 col lt  Dim col                          MatrixPiv row  col - Coefficient MatrixPiv pTarget  col                       return MatrixPiv      Determinant of Square Matrix int dCounter dRow  double Det  double MatrixDet 9  9   double Determinant double MatrixInput 9  9  int Dim        CopyMatrix MatrixInput MatrixDet Dim Dim       Det 1 0      if Dim gt 1                for dRow 1 dRow lt Dim dRow                          dCounter dRow              while  MatrixDet dRow  dRow   0 0  amp  dCounter lt  Dim                                 dCounter                    Det  -1 0                  CopyPointer InterchangeRow MatrixDet Dim Dim dRow dCounter  MatrixDet Dim Dim                             if MatrixDet dRow  dRow   0                                Det 0 0                  break                            else                               Det  MatrixDet dRow  dRow                   CopyPointer Pivote MatrixDet Dim dRow  MatrixDet Dim Dim                                   Det  MatrixDet Dim  Dim             else Det MatrixDet 1  1       return Det      Matrix Identity double MatrixIdent 9  9   double    Identity  int Dim   9        for row 1 row lt  Dim row            for col 1 col lt  Dim col                if row  col                  MatrixIdent row  col  1 0              else                 MatrixIdent row  col  0 0      return MatrixIdent      Join Matrix to be Augmented Matrix double MatrixJoin 9  9   double    JoinMatrix  double MatrixA 9  9  double MatrixB 9  9  int Row int ColA int ColB   9        Col ColA ColB      for row 1 row lt  Row row            for col 1 col lt  Col col                if col lt  ColA                  MatrixJoin row  col  MatrixA row  col               else                 MatrixJoin row  col  MatrixB row  col-ColA       return MatrixJoin      Inverse of Matrix double   Pointer  9   double IdentMatrix 9  9   int Counter  double MatrixAug 9  9   double MatrixInv 9  9   double    Inverse  double MatrixInput 9  9  int Dim   9        Row Dim      Col Dim Dim      Pointer Identity Dim       CopyPointer Pointer IdentMatrix Dim Dim       Pointer JoinMatrix MatrixInput IdentMatrix Dim Dim Dim       CopyPointer Pointer MatrixAug Row Col       for Counter 1 Counter lt  Dim Counter                     Pointer PivoteDown MatrixAug Row Col Counter Counter           CopyPointer Pointer MatrixAug Row Col             for Counter Dim Counter gt 1 Counter--                Pointer PivoteUp MatrixAug Row Col Counter Counter           CopyPointer Pointer MatrixAug Row Col             for row 1 row lt  Dim row            for col 1 col lt  Dim col                MatrixInv row  col  MatrixAug row  col Dim       return MatrixInv      Gauss-Jordan Elemination double MatrixGJ 9  9   double VectorGJ 9  9   double    GaussJordan  double MatrixInput 9  9  double VectorInput 9  9  int Dim   9        Row Dim      Col Dim 1      Pointer JoinMatrix MatrixInput VectorInput Dim Dim 1       CopyPointer Pointer MatrixGJ Row Col       for Counter 1 Counter lt  Dim Counter                     Pointer PivoteDown MatrixGJ Row Col Counter Counter           CopyPointer Pointer MatrixGJ Row Col             for Counter Dim Counter gt 1 Counter--                Pointer PivoteUp MatrixGJ Row Col Counter Counter           CopyPointer Pointer MatrixGJ Row Col             for row 1 row lt  Dim row            for col 1 col lt  1 col                VectorGJ row  col  MatrixGJ row  col Dim       return VectorGJ      Generalized Gauss-Jordan Elemination double MatrixGGJ 9  9   double VectorGGJ 9  9   double    GeneralizedGaussJordan  double MatrixInput 9  9  double VectorInput 9  9  int Dim int vCol   9        Row Dim      Col Dim vCol      Pointer JoinMatrix MatrixInput VectorInput Dim Dim vCol       CopyPointer Pointer MatrixGGJ Row Col       for Counter 1 Counter lt  Dim Counter                     Pointer PivoteDown MatrixGGJ Row Col Counter Counter           CopyPointer Pointer MatrixGGJ Row Col             for Counter Dim Counter gt 1 Counter--                Pointer PivoteUp MatrixGGJ Row Col Counter Counter           CopyPointer Pointer MatrixGGJ Row Col             for row 1 row lt  Row row            for col 1 col lt  vCol col                VectorGGJ row  col  MatrixGGJ row  col Dim       return VectorGGJ      Matrix Sparse  Three Diagonal Non-Zero Elements double MatrixSpa 9  9   double    Sparse  int Dimension double FirstElement double SecondElement double ThirdElement   9        MatrixSpa 1  1  SecondElement      MatrixSpa 1  2  ThirdElement      MatrixSpa Dimension  Dimension-1  FirstElement      MatrixSpa Dimension  Dimension  SecondElement      for int Counter 2 Counter lt Dimension Counter                  MatrixSpa Counter  Counter-1  FirstElement          MatrixSpa Counter  Counter  SecondElement          MatrixSpa Counter  Counter 1  ThirdElement            return MatrixSpa      Copy and save the above code as Matrix h then try the following code    include lt iostream gt   include lt conio h gt   include Matrix h  int Dim  double Matrix 9  9   int main         cout lt  lt  Enter your matrix dimension         cin gt  gt Dim      Input Matrix Dim Dim       cout lt  lt  Your matrix   lt  lt endl      Output Matrix Dim Dim       cout lt  lt  The inverse   lt  lt endl      Output Inverse Matrix Dim  Dim Dim       getch

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Function for inverse of the input square matrix  J  of dimension  dim    vector lt vector lt double  gt   gt  inverseVec33 vector lt vector lt double  gt   gt  J  int dim      Matrix of Minors  vector lt vector lt double  gt   gt  invJ dim vector lt double  gt   dim    for int i 0  i lt dim  i          for int j 0  j lt dim  j                  invJ i  j     J  i 1  dim   j 1  dim  J  i 2  dim   j 2  dim  -                       J  i 2  dim   j 1  dim  J  i 1  dim   j 2  dim               determinant of the matrix  double detJ   0 0  for int j 0  j lt dim  j      detJ    J 0  j  invJ 0  j       Inverse of the given matrix   vector lt vector lt double  gt   gt  invJT dim vector lt double  gt   dim     for int i 0  i lt dim  i          for int j 0  j lt dim  j                  invJT i  j    invJ j  i  detJ           return invJT     void main           given matrix  vector lt vector lt double  gt   gt  Jac 3 vector lt double  gt   3    Jac 0  0    1  Jac 0  1    2   Jac 0  2    6  Jac 1  0    -3  Jac 1  1    4   Jac 1  2    3  Jac 2  0    5  Jac 2  1    1   Jac 2  2    -4      Inverse of the matrix Jac  vector lt vector lt double  gt   gt  JacI 3 vector lt double  gt   3          call function and store inverse of J as JacI  JacI   inverseVec33 Jac 3

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I have just created a QMatrix class  It uses the built in vector   container  QMatrix h It uses the Jordan-Gauss method to compute the inverse of a square matrix   You can use it as follows    include  QMatrix h   include  lt iostream gt   int main    QMatrix lt double gt  A 3 3 true   QMatrix lt double gt  Result   A inverse   A    should give the idendity matrix  std  cout lt  lt A inverse   lt  lt std  endl  std  cout lt  lt Result lt  lt std  endl     for checking return 0      The inverse function is implemented as follows   Given a class with the following fields   template lt class T gt  class QMatrix  public  int rows  cols  std  vector lt std  vector lt T gt   gt  A    the inverse   function   template lt class T gt   QMatrix lt T gt  QMatrix lt T gt    inverse    Identity lt T gt  Id rows     the Identity Matrix as a subclass of QMatrix  QMatrix lt T gt  Result    this     making a copy and transforming it to the Identity matrix T epsilon   0 000001  for int i 0 i lt rows   i         check if Result i i   0  if true  switch the row with another      for int j i j lt rows   j           if std  abs Result j j   lt epsilon      uses Overloading   int int  to extract element from Result Matrix             Result replace rows i j 1     switches rows i with j 1                   else break               main part  making a triangular matrix     Id i  Id i   1 0 Result i i        Result i  Result i   1 0 Result i i        Using overloading    int  to get a row form the matrix     for int j i 1 j lt rows   j           T temp   Result j i           Result j    Result j  - Result i  temp          Id j    Id j  - Id i  temp    doing the same operations to the identity matrix         Result j i  0    not necessary  but looks nicer than 10 -15             solving a triangular matrix  for int i rows-1 i gt 0 --i       for int j i-1 j gt  0 --j           T temp   Result j i           Id j    Id j  - temp Id i           Result j  Result j -temp Result i            return Id

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With all due respect to our unknown  yahoo  poster  I look at code like that and just die a little inside   Alphabet soup is just so insanely difficult to debug   A single typo anywhere in there can really ruin your whole day   Sadly  this particular example lacked variables with underscores   It s so much more fun when we have  a b-c d e f-g h   Especially when using a font where   and - have the same pixel length   Taking up Suvesh Pratapa on his suggestion  I note   Given 3x3 matrix         y0x0  y0x1  y0x2        y1x0  y1x1  y1x2        y2x0  y2x1  y2x2 Declared as double matrix    Y   3     X   3      A  When taking a minor of a 3x3 array  we have 4 values of interest  The lower X Y index is always 0 or 1   The higher X Y index is always 1 or 2   Always   Therefore   double determinantOfMinor  int          theRowHeightY                             int          theColumnWidthX                             const double theMatrix    Y   3     X   3        int x1   theColumnWidthX    0   1   0      always either 0 or 1      int x2   theColumnWidthX    2   1   2      always either 1 or 2      int y1   theRowHeightY      0   1   0      always either 0 or 1      int y2   theRowHeightY      2   1   2      always either 1 or 2       return   theMatrix  y1   x1      theMatrix  y2   x2          -    theMatrix  y1   x2      theMatrix  y2   x1          B  Determinant is now    Note the minus sign    double determinant  const double theMatrix    Y   3     X   3        return   theMatrix  0   0      determinantOfMinor  0  0  theMatrix           -    theMatrix  0   1      determinantOfMinor  0  1  theMatrix                theMatrix  0   2      determinantOfMinor  0  2  theMatrix           C  And the inverse is now   bool inverse  const double theMatrix    Y   3     X   3                       double theOutput    Y   3     X   3        double det   determinant  theMatrix            Arbitrary for now   This should be something nicer         if   ABS det   lt  1e-2           memset  theOutput  0  sizeof theOutput        return false         double oneOverDeterminant   1 0   det     for     int y   0   y  lt  3   y          for   int x   0   x  lt  3   x                         Rule is inverse   1 det   minor of the TRANSPOSE matrix                Note  y x  becomes  x y  INTENTIONALLY here                        theOutput  y   x            determinantOfMinor  x  y  theMatrix     oneOverDeterminant               y0 x1    y1 x0    y1 x2   and  y2 x1   all need to be negated           if  1      x   y    2            theOutput  y   x    - theOutput  y   x            return true      And round it out with a little lower-quality testing code   void printMatrix  const double theMatrix    Y   3     X   3        for   int y   0   y  lt  3   y              cout  lt  lt             for   int x   0   x  lt  3   x              cout  lt  lt  theMatrix  y   x   lt  lt            cout  lt  lt       lt  lt  endl        cout  lt  lt  endl     void matrixMultiply   const double theMatrixA    Y   3     X   3                         const double theMatrixB    Y   3     X   3                               double theOutput     Y   3     X   3         for     int y   0   y  lt  3   y          for   int x   0   x  lt  3   x                    theOutput  y   x    0        for   int i   0   i  lt  3   i              theOutput  y   x      theMatrixA  y   i    theMatrixB  i   x            int main int argc  char   argv      if   argc  gt  1       SRANDOM  atoi  argv 1          double m 3  3        RANDOM D 0 1e3   RANDOM D 0 1e3   RANDOM D 0 1e3                            RANDOM D 0 1e3   RANDOM D 0 1e3   RANDOM D 0 1e3                            RANDOM D 0 1e3   RANDOM D 0 1e3   RANDOM D 0 1e3         double o 3  3   mm 3  3      if   argc  lt   2       cout  lt  lt  fixed  lt  lt  setprecision 3      printMatrix m     cout  lt  lt  endl  lt  lt  endl     SHOW  determinant m       cout  lt  lt  endl  lt  lt  endl     BOUT  inverse m  o       printMatrix m     printMatrix o     cout  lt  lt  endl  lt  lt  endl     matrixMultiply  m  o  mm      printMatrix m     printMatrix o     printMatrix mm       cout  lt  lt  endl  lt  lt  endl        Afterthought   You may also want to detect very large determinants as round-off errors will affect your accuracy

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Why don t you try to code it yourself  Take it as a challenge      For a 3  3 matrix   source  wolfram com     the matrix inverse is   source  wolfram com     I m assuming you know what the determinant of a matrix  A  is      Images  c  Wolfram Alpha and   mathworld wolfram  06-11-09    22 06

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Here s a version of batty s answer  but this computes the correct inverse  batty s version computes the transpose of the inverse      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     double invdet   1   det   Matrix33d minv     inverse of matrix m minv 0  0     m 1  1    m 2  2  - m 2  1    m 1  2     invdet  minv 0  1     m 0  2    m 2  1  - m 0  1    m 2  2     invdet  minv 0  2     m 0  1    m 1  2  - m 0  2    m 1  1     invdet  minv 1  0     m 1  2    m 2  0  - m 1  0    m 2  2     invdet  minv 1  1     m 0  0    m 2  2  - m 0  2    m 2  0     invdet  minv 1  2     m 1  0    m 0  2  - m 0  0    m 1  2     invdet  minv 2  0     m 1  0    m 2  1  - m 2  0    m 1  1     invdet  minv 2  1     m 2  0    m 0  1  - m 0  0    m 2  1     invdet  minv 2  2     m 0  0    m 1  1  - m 1  0    m 0  1     invdet

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include  lt conio h gt    include lt iostream h gt   const int size   9   int main         char ch       do               clrscr            int i  j  x  y  z  det  a size   b size            cout  lt  lt                   MATRIX OF 3x3 ORDER                     lt  lt  endl               lt  lt  endl               lt  lt  endl           for  i   0  i  lt   size  i                a i  0           for  i   0  i  lt  size  i                          cout  lt  lt   Enter                     lt  lt  i   1                   lt  lt    element of matrix                 cin  gt  gt  a i                 cout  lt  lt  endl                   lt  lt endl                     clrscr             cout  lt  lt   your entered matrix is                 lt  lt  endl               lt  lt endl           for  i   0  i  lt  size  i    3              cout  lt  lt  a i                    lt  lt                         lt  lt  a i 1                    lt  lt                         lt  lt  a i 2                    lt  lt  endl                   lt  lt endl           cout  lt  lt   Transpose of given matrix is                lt  lt  endl               lt  lt  endl           for  i   0  i  lt  3  i                cout  lt  lt  a i                    lt  lt                         lt  lt  a i 3                    lt  lt                         lt  lt  a i 6                    lt  lt  endl                   lt  lt  endl           cout  lt  lt   Determinent of given matrix is               x   a 0     a 4    a 8  -a  5    a 7            y   a 1     a 3    a 8  -a  5    a 6            z   a 2     a 3    a 7  -a  4    a 6            det   x - y   z           cout  lt  lt  det                lt  lt  endl               lt  lt  endl               lt  lt  endl               lt  lt  endl           if  det    0                        cout  lt  lt   As Determinent 0 so it is singular matrix and its inverse cannot exist                    lt  lt  endl                   lt  lt  endl               goto quit                     b 0    a 4    a 8  - a 5    a 7           b 1    a 5    a 6  - a 3    a 8           b 2    a 3    a 7  - a 4    a 6           b 3    a 2    a 7  - a 1    a 8           b 4    a 0    a 8  - a 2    a 6           b 5    a 1    a 6  - a 0    a 7           b 6    a 1    a 5  - a 2    a 4           b 7    a 2    a 3  - a 0    a 5           b 8    a 0    a 4  - a 1    a 3            cout  lt  lt   Adjoint of given matrix is                lt  lt  endl               lt  lt  endl           for  i   0  i  lt  3  i                          cout  lt  lt  b i                    lt  lt                         lt  lt  b i 3                    lt  lt                         lt  lt  b i 6                    lt  lt  endl                   lt  lt endl                     cout  lt  lt  endl               lt  lt endl           cout  lt  lt   Inverse of given matrix is                 lt  lt  endl               lt  lt  endl               lt  lt  endl           for  i   0  i  lt  3  i                          cout  lt  lt  b i                    lt  lt                        lt  lt  det                   lt  lt                         lt  lt  b i 3                    lt  lt                         lt  lt  det                   lt  lt                         lt  lt  b i 6                    lt  lt                         lt  lt  det                   lt  lt  endl                    lt  lt endl                     quit           cout  lt  lt  endl               lt  lt  endl           cout  lt  lt   Do You want to continue this again press  y yes n no             cin  gt  gt  ch            cout  lt  lt  endl               lt  lt  endl           end do         while  ch     y        getch          return 0

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I went ahead and wrote it in python since I think it s a lot more readable than in c   for a problem like this  The function order is in order of operations for solving this by hand via this video  Just import this and call  print invert  on your matrix    def print invert  matrix     i matrix   invert matrix  matrix    for line in i matrix      print  line    return  def invert matrix  matrix     determinant   str  determinant of 3x3  matrix     cofactor   make cofactor  matrix    trans matrix   transpose cofactor  cofactor     trans matrix        str  element        determinant for element in row  for row in trans matrix     return trans matrix  def determinant of 3x3  matrix     multiplication   1   neg multiplication   1   total   0   for start column in range  3       for row in range  3         multiplication    matrix row   start column row  3        neg multiplication    matrix row   start column-row  3      total    multiplication - neg multiplication     multiplication   neg multiplication   1   if total    0      total   1   return total  def make cofactor  matrix     cofactor     0 0 0   0 0 0   0 0 0     matrix 2x2     0 0   0 0       For each element in matrix      for row in range  3       for column in range  3               make the 2x2 matrix in this inner loop       matrix 2x2   make 2x2 from spot in 3x3  row  column  matrix        cofactor row  column    determinant of 2x2  matrix 2x2     return flip signs  cofactor   def make 2x2 from spot in 3x3  row  column  matrix     c count   0   r count   0   matrix 2x2     0 0   0 0          make the 2x2 matrix in this inner loop   for inner row in range  3       for inner column in range  3         if row is not inner row and inner column is not column          matrix 2x2 r count   2  c count   2    matrix inner row  inner column          c count    1     if row is not inner row        r count    1   return matrix 2x2  def determinant of 2x2  matrix     total   matrix 0  0    matrix  1  1    return total -  matrix  1  0    matrix  0  1    def flip signs  cofactor     sign pos   True      For each element in matrix      for row in range  3       for column in range  3         if sign pos          sign pos   False       else          cofactor row  column     -1         sign pos   True   return cofactor  def transpose cofactor  cofactor     new cofactor     0 0 0   0 0 0   0 0 0     for row in range  3       for column in range  3         new cofactor column  row    cofactor row  column    return new cofactor

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include  lt iostream gt  using namespace std   int main         double A11  A12  A13      double A21  A22  A23      double A31  A32  A33       double B11  B12  B13      double B21  B22  B23      double B31  B32  B33       cout  lt  lt   Enter all number from left to right  from top to bottom  and press enter after every number         cin   gt  gt  A11      cin   gt  gt  A12      cin   gt  gt  A13      cin   gt  gt  A21      cin   gt  gt  A22      cin   gt  gt  A23      cin   gt  gt  A31      cin   gt  gt  A32      cin   gt  gt  A33       B11   1     A22   A33  -  A23   A32        B12   1     A13   A32  -  A12   A33        B13   1     A12   A23  -  A13   A22        B21   1     A23   A31  -  A21   A33        B22   1     A11   A33  -  A13   A31        B23   1     A13   A21  -  A11   A23        B31   1     A21   A32  -  A22   A31        B32   1     A12   A31  -  A11   A32        B33   1     A11   A22  -  A12   A21         cout  lt  lt  B11  lt  lt    t   lt  lt  B12  lt  lt    t   lt  lt  B13  lt  lt  endl      cout  lt  lt  B21  lt  lt    t   lt  lt  B22  lt  lt    t   lt  lt  B23  lt  lt  endl      cout  lt  lt  B31  lt  lt    t   lt  lt  B32  lt  lt    t   lt  lt  B33  lt  lt  endl       return 0

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Don t try to do this yourself if you re serious about getting edge cases right  So while they many naive simple methods are theoretically exact  they can have nasty numerical behavior for nearly singular matrices  In particular you can get cancelation round-off errors that cause you to get arbitrarily bad results    A  correct  way is Gaussian elimination with row and column pivoting so that you re always dividing by the largest remaining numerical value   This is also stable for NxN matrices    Note that row pivoting alone doesn t catch all the bad cases   However IMO implementing this right and fast is not worth your time - use a well tested library and there are a heap of header only ones

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A rather nice  I think  header file containing macros for most 2x2  3x3 and 4x4 matrix operations has been available with most OpenGL toolkits  Not as standard but I ve seen it at various places   You can check it out here  At the end of it you will find both inverse of 2x2  3x3 and 4x4      vvector h

[c++] Simple 3x3 matrix inverse code (C++)

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