18 Cofactors

The determinant of the submatrix of $A$ that excludes row $j$ and column $k$, multiplied by ${(-1)}^{j+k}$, is called the cofactor $A_{jk}$ of $a_{jk}$, so

is the expansion by column $k$ for all $k=1,\mathrm{\dots },n$. The adjugate matrix $\text{adj}(A)$ is the transpose of the matrix $[A_{jk}]$ of cofactors.

(i) For all square matrices $A\text{adj}(A)=(detA)I$.

(ii) A square matrix is said to be lower triangular if all the entries above the leading diagonal are zero. The determinant of a lower triangular matrix equals the product of the entries on the leading diagonal.