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 C

Answer 1

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) < 1e-2 )
  {
    memset( theOutput, 0, sizeof theOutput );
    return false;
  }

  double oneOverDeterminant = 1.0 / det;

  for (   int y = 0;  y < 3;  y ++ )
    for ( int x = 0;  x < 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 < 3;  y ++ )
  {
    cout << "[  ";
    for ( int x = 0;  x < 3;  x ++   )
      cout << theMatrix [y] [x] << "  ";
    cout << "]" << endl;
  }
  cout << 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 < 3;  y ++ )
    for ( int x = 0;  x < 3;  x ++   )
    {
      theOutput [y] [x] = 0;
      for ( int i = 0;  i < 3;  i ++ )
        theOutput [y] [x] +=  theMatrixA [y] [i] * theMatrixB [i] [x];
    }
}

int
main(int argc, char **argv)
{
  if ( argc > 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 <= 2 )
    cout << fixed << setprecision(3);

  printMatrix(m);
  cout << endl << endl;

  SHOW( determinant(m) );
  cout << endl << endl;

  BOUT( inverse(m, o) );
  printMatrix(m);
  printMatrix(o);
  cout << endl << endl;

  matrixMultiply (m, o, mm );
  printMatrix(m);
  printMatrix(o);
  printMatrix(mm);  
  cout << endl << endl;
}

---

Afterthought:

You may also want to detect very large determinants as round-off errors will affect your accuracy!

Answer 2

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;

Answer 3

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)

Answer 4

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.

Answer 5

//Function for inverse of the input square matrix 'J' of dimension 'dim':

vector<vector<double > > inverseVec33(vector<vector<double > > J, int dim)
{
//Matrix of Minors
 vector<vector<double > > invJ(dim,vector<double > (dim));
for(int i=0; i<dim; i++)
{
    for(int j=0; j<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<dim; j++)
{ detJ += J[0][j]*invJ[0][j];}

//Inverse of the given matrix.
 vector<vector<double > > invJT(dim,vector<double > (dim));
 for(int i=0; i<dim; i++)
{
    for(int j=0; j<dim; j++)
    {
        invJT[i][j] = invJ[j][i]/detJ;
    }
}

return invJT;
}

void main()
{
    //given matrix:
vector<vector<double > > Jac(3,vector<double > (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<vector<double > > JacI(3,vector<double > (3));
    //call function and store inverse of J as JacI:
JacI = inverseVec33(Jac,3);
}

Answer 6

# include <conio.h>
# include<iostream.h>

const int size = 9;

int main()
{
    char ch;

    do
    {
        clrscr();
        int i, j, x, y, z, det, a[size], b[size];

        cout << "           **** MATRIX OF 3x3 ORDER ****"
             << endl
             << endl
             << endl;

        for (i = 0; i <= size; i++)
            a[i]=0;

        for (i = 0; i < size; i++)
        {
            cout << "Enter "
                 << i + 1
                 << " element of matrix=";

            cin >> a[i]; 

            cout << endl
                 <<endl;
        }

        clrscr();

        cout << "your entered matrix is "
             << endl
             <<endl;

        for (i = 0; i < size; i += 3)
            cout << a[i]
                 << "  "
                 << a[i+1]
                 << "  "
                 << a[i+2]
                 << endl
                 <<endl;

        cout << "Transpose of given matrix is"
             << endl
             << endl;

        for (i = 0; i < 3; i++)
            cout << a[i]
                 << "  "
                 << a[i+3]
                 << "  "
                 << a[i+6]
                 << endl
                 << endl;

        cout << "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 << det 
             << endl
             << endl
             << endl
             << endl;

        if (det == 0)
        {
            cout << "As Determinent=0 so it is singular matrix and its inverse cannot exist"
                 << endl
                 << 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 << "Adjoint of given matrix is"
             << endl
             << endl;

        for (i = 0; i < 3; i++)
        {
            cout << b[i]
                 << "  "
                 << b[i+3]
                 << "  "
                 << b[i+6]
                 << endl
                 <<endl;
        }

        cout << endl
             <<endl;

        cout << "Inverse of given matrix is "
             << endl
             << endl
             << endl;

        for (i = 0; i < 3; i++)
        {
            cout << b[i]
                 << "/"
                 << det
                 << "  "
                 << b[i+3]
                 << "/" 
                 << det
                 << "  "
                 << b[i+6]
                 << "/" 
                 << det
                 << endl
                  <<endl;
        }

        quit:

        cout << endl
             << endl;

        cout << "Do You want to continue this again press (y/yes,n/no)";

        cin >> ch; 

        cout << endl
             << endl;
    } /* end do */

    while (ch == 'y');
    getch ();

    return 0;
}