## Quick Reference

Let **A** be the square matrix [*a*_{ij}]. The cofactor, *A*_{ij}, of the entry *a*_{ij} is equal to (−1)^{i+j} times the determinant of the matrix obtained by deleting the *i*-th row and *j*-th column of **A**. If **A** is the 3 × 3 matrix shown, the factor (−1)^{i+j} has the effect of introducing a+or−sign according to the pattern on the right: So, for example, for a 2 × 2 matrix, the pattern is:So, the cofactor of *a* equals *d*, the cofactor of *b* equals−*c*, and so on. The following properties hold, for an *n* × *n* matrix **A**: *a*_{i1}*A*_{i1}+*a*_{i2}+⋯+ *a*_{in}*A*_{in} has the same value for any *i*, and is the definition of det **A**, the determinant of **A**. This particular expression is the evaluation of det **A** by the *i*-th row.*i* ≠ *j*, *a*_{i1}*A*_{j1}+*a*_{i2}*A*_{j1}+⋯+*a*_{in}*A*_{in}=0.Results for columns, corresponding to the results (i) and (ii) for rows, also hold.

*a*_{i1}*A*_{i1}+*a*_{i2}+⋯+ *a*_{in}*A*_{in} has the same value for any *i*, and is the definition of det **A**, the determinant of **A**. This particular expression is the evaluation of det **A** by the *i*-th row.

*i* ≠ *j*, *a*_{i1}*A*_{j1}+*a*_{i2}*A*_{j1}+⋯+*a*_{in}*A*_{in}=0.

*Subjects:*
Mathematics.

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