Cofactor
\[A_{ij} = (-1)^{i+j}M_{ij}\]
\(M_{ij}\): determinant left after deleting row \(i\) and column \(j\).
Determinants
Minors and Cofactors
Understand
The minor \(M_{ij}\) of the element \(a_{ij}\) is the determinant obtained by deleting row \(i\) and column \(j\). The cofactor is
\[A_{ij} = (-1)^{i+j}M_{ij}.\]
The determinant equals the sum of the elements of any row (or column) multiplied by their own cofactors:
\[|A| = a_{11}A_{11} + a_{12}A_{12} + a_{13}A_{13}.\]
If the elements of one row are multiplied by the cofactors of a different row, the sum is always 0. This fact is the reason \(A(\operatorname{adj}A) = |A|I\).
Key Concepts
- Minor: delete row & column, evaluate.
- Cofactor: attach the sign \((-1)^{i+j}\).
Formula Bank
Common Mistakes
Sign of the cofactor
\(A_{ij}=(-1)^{i+j}M_{ij}\). The sign pattern is a chessboard starting with + in the top-left corner.
Practice & Topic Test
Topic Practice
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\(A_{ij} = (-1)^{i+j}M_{ij}\). Own cofactors ⇒ \(|A|\); another row’s cofactors ⇒ 0.
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