Matrix Adjugate

Worked Example

Consider the 3x3 matrix A:

A = [
  [ 1,  2,  0],
  [ 0,  1,  1],
  [ 1,  0,  1]
]

First, we compute the determinant of A:

det(A) = 1*(1*1 - 1*0) - 2*(0*1 - 1*1) + 0 = 1(1) - 2(-1) = 3

Next, we compute the cofactor for the entry in row 1, column 2 (A₁₂ = 2). We delete row 1 and column 2 to get the minor matrix M₁₂:

M_12 = [
  [ 0, 1 ],
  [ 1, 1 ]
]

The determinant of this minor is (0*1 - 1*1) = -1. The sign for position (1, 2) is (-1)¹⁺² = -1. Therefore, the cofactor C₁₂ = -1 * (-1) = 1.

After computing all cofactors, we get the cofactor matrix C:

C = [
  [  1,  1, -1 ],
  [ -2,  1,  2 ],
  [  2, -1,  1 ]
]

Finally, the adjugate of A, denoted adj(A), is the transpose of C (C^T):

adj(A) = [
  [  1, -2,  2 ],
  [  1,  1, -1 ],
  [ -1,  2,  1 ]
]

Notice that A · adj(A) = det(A) · I = 3I.

Experiment

Try the Matrix Adjugate interactive module to explore how changes in matrix entries affect the cofactor matrix and the adjugate.

1. Set the values of A11 to A33 to create a non-symmetric integer matrix (e.g. A11=1, A12=2, A22=1, A23=1, A31=1, A33=1, rest 0).
2. Adjust the Cofactor row and Cofactor column controls to view how the selected minor and its corresponding cofactor value are derived.
3. Observe that the cofactor computed at position (i, j) appears at position (j, i) in the final adjugate matrix, verifying the transpose step.
4. Set the matrix entries so the determinant becomes zero (e.g., duplicate two rows). Notice that the adjugate might still be non-zero (rank 1), demonstrating that A · adj(A) = 0 · I = 0 matrix.

Limitations

This module is designed for 3x3 integer matrices to keep manual computation and visualization manageable. The concepts of minors, cofactors, and adjugate matrices generalize to any n x n matrix. However, calculating determinants and adjugates algebraically becomes computationally expensive for larger matrices, having a time complexity of O(n!). In practical applications, algorithms like LU decomposition are used to compute inverses and determinants more efficiently.

References

• SymPy Documentation: Matrices

The SymPy mathematical library documentation offers a rigorous view on matrix algebraic operations, including the calculation of determinants, cofactors, and classical adjugate matrices. It provides software-based implementations reflecting the exact theoretical formulas shown in this module.