Let A be a n × n matrix such that A is upper-triangular. Then adj A =

Let, ${\left[A\right]}_{n\times n} =\left[\begin{array}{cccc}{a}_{11}& a_{12}& a_{13}........& a_{1n}\\ 0& a_{22}& a_{23}......& a_{2n}\\ 0& 0& a_{33........}& a_{3n}\\ ......& ....& .......& .....\\ 0& 0& 0& a_{nn}\end{array}\right]$

Cofactors of all the elements of upper triangle are zero.

$\mathrm{adj}\left(A\right)={\left[\begin{array}{cccc}{A}_{11}& 0& 0........& 0\\ A_{12}& A_{22}& 0......& 0\\ A_{13}& A_{23}& A_{33 ........}& 0\\ ......& ....& .......& .....\\ A_{1n}& A_{2n}& .....& A_{nn}\end{array}\right]}^T$

$\mathrm{adj}\left(A\right)=\left[\begin{array}{cccc}{A}_{11}& A_{12}& A_{13}........& A_{1n}\\ 0& A_{22}& A_{23}......& A_{2n}\\ 0& 0& A_{33........}& A_{3n}\\ ......& ....& .......& .....\\ 0& 0& 0& A_{nn}\end{array}\right]$

Hence, $\mathrm{adj}\left(A\right)$ is also upper triangular matrix