Connection between adjoint of a matrix and adjoint of an operator

Question

Let $T:\mathbb{R}^2 \rightarrow \mathbb{R}^2$ with $$T(x,y) = \left[ \begin{array}{ccc} 1x+2y \\ 3x+4y \end{array} \right] $$

The matrix representation of $T$ is $$ A= \left[ \begin{array}{ccc} 1 & 2 \\ 3 & 4 \end{array} \right].$$

Now the adjoint of the operator $T$ is the transpose of $A$ $$ \left[ \begin{array}{ccc} 1 & 3 \\ 2 & 4 \end{array} \right],$$

"Adjoints of operators generalize (conjugate) transposes of square matrices."

On the other hand, the adjoint of the matrix $A$ is $$\left[ \begin{array}{ccc} 4 & -2 \\ -3 & 1 \end{array} \right].$$

Is there any connection between the two? or the two "adjoint" definitions used here are unrelated.

Thanks!

Answer 1

I believe you are mistaken. The adjoint of the matrix A is the transpose of the matrix A.

One major confusion here is that there are two definitions for the word adjoint. The adjoint of a matrix is its conjugate transpose. Another definition, now often called the "classical adjoint" of a matrix is the matrix of its cofactors, which is what I think you write as the adjoint of the matrix A. Another now more common name for the classical adjoint of a matrix is its adjugate.