Computing Determinants via Cofactor Expansion
Cofactor expansion reduces the determinant of an n-by-n matrix into a weighted sum of smaller submatrix determinants.
Submatrices and Minors
To calculate the determinant of a large matrix, we break it down using smaller submatrices. The minor M_{ij} is the determinant of the (n-1)-by-(n-1) submatrix formed by deleting row i and column j from matrix A. Minors isolate the geometric scale contribution of individual entries.
M_{ij} = \det(A_{ij})Cofactors and Sign Grids
A cofactor attaches an alternating sign multiplier to a minor based on its position in the matrix grid. The sign follows a checkerboard pattern determined by raising negative one to the sum of the row and column indices. Combining an entry with its cofactor prepares it for the full expansion sum.
C_{ij} = (-1)^{i+j} M_{ij}Laplace Expansion Rule
The determinant of matrix A equals the sum of entries in any single row or column multiplied by their corresponding cofactors. Choosing a row or column with many zero entries drastically reduces the required calculations. This recursive technique allows computation of determinants for square matrices of any dimension.
\det(A) = \sum_{j=1}^{n} a_{ij} C_{ij}Flashcard
The minor corresponding to matrix entry a_{ij}
Check your understanding
For a 3x3 matrix where the second row is [0, 5, 0], which method minimizes arithmetic when computing the determinant?