Matrix Adjugate

The adjugate (sometimes called classical adjoint) of a matrix is formed by taking the transpose of its cofactor matrix. It is a key component in the formula for finding the inverse of a matrix, giving us a way to compute the inverse in a closed-form algebraic manner without requiring row reduction. Unlike the true matrix inverse, the adjugate is well-defined for all square matrices, even singular ones where the determinant is zero.

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.