Matrix Determinant

Definition

• A number that can be calculated from a matrices elements.
• Denoted: $det(A)$ or $|A|$.
• The determinant helps us find the inverse of a matrix.

Properties:

$$\begin{array}{rl}& \text{Let }A,B\in {\mathbb{R}}^{n\times n}.\text{ Then}\\ & |A|\ne 0\iff A\text{ is invertible.}\\ & \text{If }A\text{ is invertible, then }|A^{-1}|=\frac{1}{|A|}.\\ & \text{If any column or row of }A\text{ consists only of zeros, then }|A|=0.\\ & |AB|=|A||B|.\\ & |A^T|=|A|.\\ & |I|=1.\end{array}$$

• A matrix with a zero determinant indicates the system does not have a unique solution; meaning there may be no solution or infinitely many solutions.
• In game development, an invertible matrix is often necessary, especially for operations like camera transformations, where the inverse of a transformation is frequently used.

Example for a 2x2 Matrix:

$$A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$$ $$|A|=ad-bc$$

calculateDeterminant2x2(double a, double b, double c, double d) {
return (a * d) - (b * c); 
} 
 
int main() { 
double a = 1, b = 2, c = 3, d = 4; 									  
cout << "Determinant: " << calculateDeterminant2x2(a, b, c, d) << endl; return 0; 
}

Example for a 3x3 Matrix:

$$A=\left[\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right]$$ $$|A|=a(ei-fh)-b(di-fg)+c(dh-eg)$$

double calculateDeterminant(const vector<vector<double>>& matrix) {
  int n = matrix.size();
  double det = 0;
 
  // Base case for 1x1 matrix
  if (n == 1) {
      return matrix[0][0];
  }
 
  // Base case for 2x2 matrix
  if (n == 2) {
      return (matrix[0][0] * matrix[1][1]) - (matrix[0][1] * matrix[1][0]);
  }
 
  // Recursive case for matrices larger than 2x2
  int sign = 1; // Sign of cofactor
  for (int i = 0; i < n; i++) {
      // Generate minor matrix excluding current row and column
      vector<vector<double>> minorMatrix(n - 1, vector<double>(n - 1));
      for (int j = 1; j < n; j++) {
          int colIndex = 0;
          for (int k = 0; k < n; k++) {
              if (k != i) {
                  minorMatrix[j - 1][colIndex] = matrix[j][k];
                  colIndex++;
              }
          }
      }
 
      // Recursive call to calculate determinant of minor
      det += sign * matrix[0][i] * calculateDeterminant(minorMatrix);
      sign = -sign; // Alternate sign for cofactor
  }
 
  return det;
}
 
int main() {
  // Example for a 3x3 matrix
  vector<vector<double>> matrix = {
      {6, 1, 1},
      {4, -2, 5},
      {2, 8, 7}
  };
 
  cout << "Determinant of the matrix: " << calculateDeterminant(matrix) << endl;
 
  return 0;
}

Example:

$$\begin{array}{rl}A& =\left[\begin{array}{ccc}4& 3& 2\\ 1& 5& 7\\ 6& 8& 9\end{array}\right]\\ |A|& =4(5\times 9-7\times 8)-3(1\times 9-7\times 6)+2(1\times 8-5\times 6)\\ & =4(45-56)-3(9-42)+2(8-30)\\ & =4(-11)-3(-33)+2(-22)\\ & =11\end{array}$$

Example for a 4x4 Matrix:

$$A=\left[\begin{array}{cccc}a& b& c& d\\ e& f& g& h\\ i& j& k& l\\ m& n& o& p\end{array}\right]$$

• Let’s use the Laplace expansion along the first row:

$$|A|=a\cdot \left|\begin{array}{ccc}f& g& h\\ j& k& l\\ n& o& p\end{array}\right|-b\cdot \left|\begin{array}{ccc}e& g& h\\ i& k& l\\ m& o& p\end{array}\right|+c\cdot \left|\begin{array}{ccc}e& f& h\\ i& j& l\\ m& n& p\end{array}\right|-d\cdot \left|\begin{array}{ccc}e& f& g\\ i& j& k\\ m& n& o\end{array}\right|$$

• Expanding each 3x3 matrix:

$$\begin{array}{rl}|A|=& a\left[f(kp-lo)-g(jp-ln)+h(jo-kn)\right]\\ & -b\left[e(kp-lo)-g(ip-lm)+h(io-km)\right]\\ & +c\left[e(jp-ln)-f(ip-lm)+h(in-jm)\right]\\ & -d\left[e(jo-kn)-f(io-km)+g(in-jm)\right]\end{array}$$

• The final value of the determinant:

$$\begin{array}{rl}|A|=& a\left[fkp-flo-gjp+gln+hjo-hkn\right]\\ & -b\left[ekp-elo-gip+glm+hio-hkm\right]\\ & +c\left[ejp-eln-fip+flm+hin-hjm\right]\\ & -d\left[ejo-ekn-fio+fkm+gin-gjm\right]\end{array}$$

Example:

$$\begin{array}{rl}B& =\left[\begin{array}{cccc}2& 3& 4& 5\\ 6& 7& 8& 9\\ 1& 2& 3& 4\\ 5& 6& 7& 8\end{array}\right]\\ |B|& =2(7\cdot 3-8\cdot 4)-3(6\cdot 3-8\cdot 1)+4(6\cdot 4-7\cdot 1)-5(7\cdot 1-6\cdot 2)\\ & =2(21-32)-3(18-8)+4(24-7)-5(7-12)\\ & =2(-11)-3(10)+4(17)-5(-5)\\ & =60\end{array}$$

Example 2:

$$\begin{array}{rl}B& =\left[\begin{array}{cccc}2& 3& 4& 5\\ 6& 7& 8& 9\\ 1& 2& 3& 4\\ 0& 0& 0& 1\end{array}\right]\\ |B|& =2(7\cdot 3-8\cdot 4)-3(6\cdot 3-8\cdot 1)+4(6\cdot 4-7\cdot 1)-5(7\cdot 1-6\cdot 2)\\ & =2(21-32)-3(18-8)+4(24-7)-5(7-12)\\ & =2(-11)-3(10)+4(17)-5(-5)\\ & =60\end{array}$$

Determinant Unaffected by Specific Last Row

The 4x4 matrix B with a last row of [0, 0, 0, 1], the determinant remains the same as with the original last row of [5, 6, 7, 8]. This occurs because:

• The zeros in the last row do not contribute to the determinant, as multiplying by zero results in zero.
• The sole non-zero element, 1, in the last position affects the determinant the same way in both matrix versions. It’s multiplied by the determinant of the 3x3 matrix formed by excluding the last row and column, which stays constant.

Relevance in Computer Graphics

This property is crucial in computer graphics, where 4x4 matrices are extensively used for operations like translation, rotation, and scaling. The [0, 0, 0, 1] row in these transformation matrices facilitates these operations without changing the determinant, thereby preserving essential geometric properties in graphical transformations.