For \(A=\begin{pmatrix}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 10\end{pmatrix}\), what is \(a_{11}C_{21}+a_{12}C_{22}+a_{13}C_{23}\) (row 1 with the cofactors of row 2)?
Multiplying the entries of one row by the cofactors of a different row always gives \(0\) (it is the determinant of a matrix with two equal rows). With the same row it would give \(\det A=-3\).
The minor \(M\) of an entry is the determinant of the submatrix left after deleting its row and column; the cofactor \(C\) is that minor with an alternating sign.
\[C_{ij}=(-1)^{i+j}M_{ij}\]
More practice problems: cofactor and laplace expansion