Cofactor Expansion Example

Linear Algebra (Matrices) · Axiom Academy

EXAMPLE Computing by Cofactor Expansion Evaluate a determinant by expanding along a strategically chosen column. Compute the determinant of the matrix below using cofactor expansion. Nice work. You evaluated a determinant by cofactor expansion and got . Expand along zeros: for any fixed row or column — pick the one with the most zeros. Column 1 had two, so two of the three terms vanished. Sign pattern: the cofactor C_ ij =(-1)^ i+j M_ ij alternates +,-,+ in a checkerboard starting with + at the top-left. Minor: M_ ij is the determinant left after deleting entry (i,j) 's row and column. Here . rule: , so (4)(5)-(1)(-2)=22 . The same idea scales up: a determinant expands into cofactors, and so on. Always expand along the row or column with the most zeros to keep the work small.

This is the written version of the interactive lesson above. See the full Linear Algebra (Matrices) course.