I have a 2 by 2 matrix as follows: $$ A:= \begin{bmatrix} d &0\\ -c& 1 \end{bmatrix} $$ where $d$ $c$ are two constant. Then the paper I am reading, page 21, after equation $(5.3)$ claiming that $$\tag 1 [A]^* A= \begin{bmatrix} d^2+c^2 &-c\\ -c& 1 \end{bmatrix} $$ but I am keeping getting $$ [A]^* A= \begin{bmatrix} d &0\\ 0& d \end{bmatrix} $$ (where by $[A]^\ast$ we denote the adjoint)
Can anybody help me to figure how could I get $(1)$ as above?
Thank you!
PS: I got $$ [A]^*= \begin{bmatrix} 1 &0\\ c& d \end{bmatrix} $$